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[Axiom-developer] 20071228.01.tpd.patch
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20071228.01.tpd.patch |
|
Date: |
Fri, 28 Dec 2007 21:10:23 -0600 |
These are newly created pages for the new hyperdoc browser.
They are the first pages in the Computer Algebra Test Suite
branch under Topics.
This is hand-written mathml, generated from scratch, so there
might be some mistakes. Further efforts will involve connecting
the pages to the supporting algebra pamphlets and developing a new
CATS set of packages for Axiom.
You'll need to check the Fonts link to ensure you have all of
the required fonts installed. There is a bug in Firefox which
shows up as a failure to stretch absolute value bars.
Tim
=======================================================================
diff --git a/changelog b/changelog
index 01b91c6..812a22f 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,9 @@
+20071228 tpd src/hyper/bookvol11 add standards compliance for gamma
+20071228 tpd src/hyper/bitmaps/gammacomplexinverse.png added
+20071228 tpd src/hyper/bitmaps/gammacomplex.png added
+20071228 tpd src/hyper/bitmaps/gammareal3.png added
+20071228 tpd src/hyper/bitmaps/psi.png added
+20071228 tpd src/hyper/bitmaps/loggamma.png added
20071225 sxw src/interp/bookvol5 fix top-level typo
20071218 acr src/algebra/axserver.spad fix lastType output re: errors
20071217 tpd src/algebra/variable.spad ignore regression test gensym (7041)
diff --git a/src/hyper/bitmaps/gammacomplex.png
b/src/hyper/bitmaps/gammacomplex.png
new file mode 100644
index 0000000..cac4afd
Binary files /dev/null and b/src/hyper/bitmaps/gammacomplex.png differ
diff --git a/src/hyper/bitmaps/gammacomplexinverse.png
b/src/hyper/bitmaps/gammacomplexinverse.png
new file mode 100644
index 0000000..2ba6b6e
Binary files /dev/null and b/src/hyper/bitmaps/gammacomplexinverse.png differ
diff --git a/src/hyper/bitmaps/gammareal3.png b/src/hyper/bitmaps/gammareal3.png
new file mode 100644
index 0000000..55be8e5
Binary files /dev/null and b/src/hyper/bitmaps/gammareal3.png differ
diff --git a/src/hyper/bitmaps/loggamma.png b/src/hyper/bitmaps/loggamma.png
new file mode 100644
index 0000000..bff7e62
Binary files /dev/null and b/src/hyper/bitmaps/loggamma.png differ
diff --git a/src/hyper/bitmaps/psi.png b/src/hyper/bitmaps/psi.png
new file mode 100644
index 0000000..275035c
Binary files /dev/null and b/src/hyper/bitmaps/psi.png differ
diff --git a/src/hyper/bookvol11.pamphlet b/src/hyper/bookvol11.pamphlet
index 295ffe0..bd004ea 100644
--- a/src/hyper/bookvol11.pamphlet
+++ b/src/hyper/bookvol11.pamphlet
@@ -414,6 +414,32 @@ PAGES=rootpage.xhtml \
aldorusersguidepage.xhtml \
foundationlibrarydocpage.xhtml \
topicspage.xhtml \
+ cats.xhtml \
+ dlmf.xhtml \
+ dlmfapproximations.xhtml \
+ dlmfasymptoticexpansions.xhtml \
+ dlmfbarnesgfunction.xhtml \
+ dlmfbetafunction.xhtml \
+ dlmfcontinuedfractions.xhtml \
+ dlmfdefinitions.xhtml \
+ dlmffunctionrelations.xhtml \
+ dlmfgraphics.xhtml \
+ dlmfinequalities.xhtml \
+ dlmfinfiniteproducts.xhtml \
+ dlmfintegrals.xhtml \
+ dlmfintegralrepresentations.xhtml \
+ dlmfmathematicalapplications.xhtml \
+ dlmfmethodsofcomputation.xhtml \
+ dlmfmultidimensionalintegral.xhtml \
+ dlmfnotation.xhtml \
+ dlmfphysicalapplications.xhtml \
+ dlmfpolygammafunctions.xhtml \
+ dlmfqgammaandbetafunctions.xhtml \
+ dlmfseriesexpansions.xhtml \
+ dlmfsums.xhtml \
+ dlmfsoftware.xhtml \
+ dlmfspecialvaluesandextrema.xhtml \
+ dlmftables.xhtml \
uglangpage.xhtml \
examplesexposedpage.xhtml \
ugsyscmdpage.xhtml \
@@ -878,7 +904,8 @@ the javascript can be added easily.
<<standard head>>=
<?xml version="1.0" encoding="UTF-8"?>
<html xmlns="http://www.w3.org/1999/xhtml";
- xmlns:xlink="http://www.w3.org/1999/xlink";>
+ xmlns:xlink="http://www.w3.org/1999/xlink";
+ xmlns:m="http://www.w3.org/1998/Math/MathML";>
<head>
<meta http-equiv="Content-Type" content="text/html" charset="us-ascii"/>
<title>Axiom Documentation</title>
@@ -1172,6 +1199,350 @@ is currently ignored.
<th align="left">Name</th>
</tr>
<tr valign="top">
+ <td>Α</td>
+ <td>913</td>
+ <td>00391</td>
+ <td>&Alpha;</td>
+ <td>greek capital letter alpha</td>
+ </tr>
+ <tr valign="top">
+ <td>Β</td>
+ <td>914</td>
+ <td>00392</td>
+ <td>&Beta;</td>
+ <td>greek capital letter beta</td>
+ </tr>
+ <tr valign="top">
+ <td>Γ</td>
+ <td>915</td>
+ <td>00393</td>
+ <td>&Gamma;</td>
+ <td>greek capital letter gamma</td>
+ </tr>
+ <tr valign="top">
+ <td>Δ</td>
+ <td>916</td>
+ <td>00394</td>
+ <td>&Delta;</td>
+ <td>greek capital letter delta</td>
+ </tr>
+ <tr valign="top">
+ <td>Ε</td>
+ <td>917</td>
+ <td>00395</td>
+ <td>&Epsilon;</td>
+ <td>greek capital letter epsilon</td>
+ </tr>
+ <tr valign="top">
+ <td>Ζ</td>
+ <td>918</td>
+ <td>00396</td>
+ <td>&Zeta;</td>
+ <td>greek capital letter zeta</td>
+ </tr>
+ <tr valign="top">
+ <td>Η</td>
+ <td>919</td>
+ <td>00397</td>
+ <td>&Eta;</td>
+ <td>greek capital letter eta</td>
+ </tr>
+ <tr valign="top">
+ <td>Θ</td>
+ <td>920</td>
+ <td>00398</td>
+ <td>&Theta;</td>
+ <td>greek capital letter theta</td>
+ </tr>
+ <tr valign="top">
+ <td>Ι</td>
+ <td>921</td>
+ <td>00399</td>
+ <td>&Iota;</td>
+ <td>greek capital letter iota</td>
+ </tr>
+ <tr valign="top">
+ <td>Κ</td>
+ <td>922</td>
+ <td>0039A</td>
+ <td>&Kappa;</td>
+ <td>greek capital letter kappa</td>
+ </tr>
+ <tr valign="top">
+ <td>Λ</td>
+ <td>923</td>
+ <td>0039B</td>
+ <td>&Lambda;</td>
+ <td>greek capital letter lambda</td>
+ </tr>
+ <tr valign="top">
+ <td>Μ</td>
+ <td>924</td>
+ <td>0039C</td>
+ <td>&Mu;</td>
+ <td>greek capital letter mu</td>
+ </tr>
+ <tr valign="top">
+ <td>Ν</td>
+ <td>925</td>
+ <td>0039D</td>
+ <td>&Nu;</td>
+ <td>greek capital letter nu</td>
+ </tr>
+ <tr valign="top">
+ <td>Ξ</td>
+ <td>926</td>
+ <td>0039E</td>
+ <td>&Xi;</td>
+ <td>greek capital letter xi</td>
+ </tr>
+ <tr valign="top">
+ <td>Ο</td>
+ <td>927</td>
+ <td>0039F</td>
+ <td>&Omicron;</td>
+ <td>greek capital letter omicron</td>
+ </tr>
+ <tr valign="top">
+ <td>Π</td>
+ <td>928</td>
+ <td>003A0</td>
+ <td>&Pi;</td>
+ <td>greek capital letter pi</td>
+ </tr>
+ <tr valign="top">
+ <td>Ρ</td>
+ <td>929</td>
+ <td>003A1</td>
+ <td>&Rho;</td>
+ <td>greek capital letter rho</td>
+ </tr>
+ <tr valign="top">
+ <td>Σ</td>
+ <td>931</td>
+ <td>003A3</td>
+ <td>&Sigma;</td>
+ <td>greek capital letter sigma</td>
+ </tr>
+ <tr valign="top">
+ <td>Τ</td>
+ <td>932</td>
+ <td>003A4</td>
+ <td>&Tau;</td>
+ <td>greek capital letter tau</td>
+ </tr>
+ <tr valign="top">
+ <td>Υ</td>
+ <td>933</td>
+ <td>003A5</td>
+ <td>&Upsilon;</td>
+ <td>greek capital letter upsilon</td>
+ </tr>
+ <tr valign="top">
+ <td>Φ</td>
+ <td>934</td>
+ <td>003A6</td>
+ <td>&Phi;</td>
+ <td>greek capital letter phi</td>
+ </tr>
+ <tr valign="top">
+ <td>Χ</td>
+ <td>935</td>
+ <td>003A7</td>
+ <td>&Chi;</td>
+ <td>greek capital letter chi</td>
+ </tr>
+ <tr valign="top">
+ <td>Ψ</td>
+ <td>936</td>
+ <td>003A8</td>
+ <td>&Psi;</td>
+ <td>greek capital letter psi</td>
+ </tr>
+ <tr valign="top">
+ <td>Ω</td>
+ <td>937</td>
+ <td>003A9</td>
+ <td>&Omega;</td>
+ <td>greek capital letter omega</td>
+ </tr>
+ <tr valign="top">
+ <td>α</td>
+ <td>945</td>
+ <td>003B1</td>
+ <td>&alpha;</td>
+ <td>greek small letter alpha</td>
+ </tr>
+ <tr valign="top">
+ <td>β</td>
+ <td>946</td>
+ <td>003B2</td>
+ <td>&beta;</td>
+ <td>greek small letter beta</td>
+ </tr>
+ <tr valign="top">
+ <td>γ</td>
+ <td>947</td>
+ <td>003B3</td>
+ <td>&gamma;</td>
+ <td>greek small letter gamma</td>
+ </tr>
+ <tr valign="top">
+ <td>δ</td>
+ <td>948</td>
+ <td>003B4</td>
+ <td>&delta;</td>
+ <td>greek small letter delta</td>
+ </tr>
+ <tr valign="top">
+ <td>ε</td>
+ <td>949</td>
+ <td>003B5</td>
+ <td>&epsilon;</td>
+ <td>greek small letter epsilon</td>
+ </tr>
+ <tr valign="top">
+ <td>ζ</td>
+ <td>950</td>
+ <td>003B6</td>
+ <td>&zeta;</td>
+ <td>greek small letter zeta</td>
+ </tr>
+ <tr valign="top">
+ <td>η</td>
+ <td>951</td>
+ <td>003B7</td>
+ <td>&eta;</td>
+ <td>greek small letter eta</td>
+ </tr>
+ <tr valign="top">
+ <td>θ</td>
+ <td>952</td>
+ <td>003B8</td>
+ <td>&theta;</td>
+ <td>greek small letter theta</td>
+ </tr>
+ <tr valign="top">
+ <td>ι</td>
+ <td>953</td>
+ <td>003B9</td>
+ <td>&iota;</td>
+ <td>greek small letter iota</td>
+ </tr>
+ <tr valign="top">
+ <td>κ</td>
+ <td>954</td>
+ <td>003BA</td>
+ <td>&kappa;</td>
+ <td>greek small letter kappa</td>
+ </tr>
+ <tr valign="top">
+ <td>λ</td>
+ <td>955</td>
+ <td>003BB</td>
+ <td>&lambda;</td>
+ <td>greek small letter lambda</td>
+ </tr>
+ <tr valign="top">
+ <td>μ</td>
+ <td>956</td>
+ <td>003BC</td>
+ <td>&mu;</td>
+ <td>greek small letter mu</td>
+ </tr>
+ <tr valign="top">
+ <td>ν</td>
+ <td>957</td>
+ <td>003BD</td>
+ <td>&nu;</td>
+ <td>greek small letter nu</td>
+ </tr>
+ <tr valign="top">
+ <td>ξ</td>
+ <td>958</td>
+ <td>003BE</td>
+ <td>&xi;</td>
+ <td>greek small letter xi</td>
+ </tr>
+ <tr valign="top">
+ <td>ο</td>
+ <td>959</td>
+ <td>003BF</td>
+ <td>&omicron;</td>
+ <td>greek small letter omicron</td>
+ </tr>
+ <tr valign="top">
+ <td>π</td>
+ <td>960</td>
+ <td>003C0</td>
+ <td>&pi;</td>
+ <td>greek small letter pi</td>
+ </tr>
+ <tr valign="top">
+ <td>ρ</td>
+ <td>961</td>
+ <td>003C1</td>
+ <td>&rho;</td>
+ <td>greek small letter rho</td>
+ </tr>
+ <tr valign="top">
+ <td>ς</td>
+ <td>962</td>
+ <td>003C2</td>
+ <td>&sigmaf;</td>
+ <td>greek small letter final sigma</td>
+ </tr>
+ <tr valign="top">
+ <td>σ</td>
+ <td>963</td>
+ <td>003C3</td>
+ <td>&sigma;</td>
+ <td>greek small letter sigma</td>
+ </tr>
+ <tr valign="top">
+ <td>τ</td>
+ <td>964</td>
+ <td>003C4</td>
+ <td>&tau;</td>
+ <td>greek small letter tau</td>
+ </tr>
+ <tr valign="top">
+ <td>υ</td>
+ <td>965</td>
+ <td>003C5</td>
+ <td>&upsilon;</td>
+ <td>greek small letter upsilon</td>
+ </tr>
+ <tr valign="top">
+ <td>φ</td>
+ <td>966</td>
+ <td>003C6</td>
+ <td>&phi;</td>
+ <td>greek small letter phi</td>
+ </tr>
+ <tr valign="top">
+ <td>χ</td>
+ <td>967</td>
+ <td>003C7</td>
+ <td>&chi;</td>
+ <td>greek small letter chi</td>
+ </tr>
+ <tr valign="top">
+ <td>ψ</td>
+ <td>968</td>
+ <td>003C8</td>
+ <td>&psi;</td>
+ <td>greek small letter psi</td>
+ </tr>
+ <tr valign="top">
+ <td>ω</td>
+ <td>969</td>
+ <td>003C9</td>
+ <td>&omega;</td>
+ <td>greek small letter omega</td>
+ </tr>
+ <tr><td>----</td><td>----</td><td>----</td><td>----</td><td>----</td></tr>
+ <tr valign="top">
<td>¯</td>
<td>175</td>
<td>000AF</td>
@@ -1179,6 +1550,20 @@ is currently ignored.
<td>macron</td>
</tr>
<tr valign="top">
+ <td>±</td>
+ <td>177</td>
+ <td>000B1</td>
+ <td>&plusmn;</td>
+ <td>plus-or-minus sign</td>
+ </tr>
+ <tr valign="top">
+ <td>×</td>
+ <td>215</td>
+ <td>000D7</td>
+ <td></td>
+ <td>multiplication sign</td>
+ </tr>
+ <tr valign="top">
<td>è</td>
<td>232</td>
<td>000E8</td>
@@ -1207,6 +1592,13 @@ is currently ignored.
<td>horizontal ellipsis</td>
</tr>
<tr valign="top">
+ <td>⋯</td>
+ <td>8943</td>
+ <td>022EF</td>
+ <td></td>
+ <td>midline horizontal ellipsis</td>
+ </tr>
+ <tr valign="top">
<td>′</td>
<td>8242</td>
<td>02032</td>
@@ -1228,6 +1620,83 @@ is currently ignored.
<td>invisible times</td>
</tr>
<tr valign="top">
+ <td>ℂ</td>
+ <td>8450</td>
+ <td>02102</td>
+ <td></td>
+ <td>doube-struck captial c</td>
+ </tr>
+ <tr valign="top">
+ <td>ℍ</td>
+ <td>8461</td>
+ <td>0210D</td>
+ <td></td>
+ <td>double-struck captial h</td>
+ </tr>
+ <tr valign="top">
+ <td>ℑ</td>
+ <td>8465</td>
+ <td>02111</td>
+ <td>&image;</td>
+ <td>black-letter captial i</td>
+ </tr>
+ <tr valign="top">
+ <td>ℓ</td>
+ <td>8467</td>
+ <td>02113</td>
+ <td></td>
+ <td>script small l</td>
+ </tr>
+ <tr valign="top">
+ <td>ℕ</td>
+ <td>8469</td>
+ <td>02115</td>
+ <td></td>
+ <td>double-struck captial n</td>
+ </tr>
+ <tr valign="top">
+ <td>ℙ</td>
+ <td>8473</td>
+ <td>02119</td>
+ <td></td>
+ <td>double-struck captial p</td>
+ </tr>
+ <tr valign="top">
+ <td>ℚ</td>
+ <td>8474</td>
+ <td>0211A</td>
+ <td></td>
+ <td>double-struck captial q</td>
+ </tr>
+ <tr valign="top">
+ <td>ℜ</td>
+ <td>8476</td>
+ <td>0211C</td>
+ <td>&real;</td>
+ <td>black-letter captial r</td>
+ </tr>
+ <tr valign="top">
+ <td>ℝ</td>
+ <td>8477</td>
+ <td>0211D</td>
+ <td></td>
+ <td>double-struck captial r</td>
+ </tr>
+ <tr valign="top">
+ <td>ℤ</td>
+ <td>8484</td>
+ <td>02124</td>
+ <td></td>
+ <td>double-struck captial z</td>
+ </tr>
+ <tr valign="top">
+ <td>ⅅ</td>
+ <td>8517</td>
+ <td>02145</td>
+ <td></td>
+ <td>doube-struck captial d</td>
+ </tr>
+ <tr valign="top">
<td>ⅆ</td>
<td>8518</td>
<td>02146</td>
@@ -3051,6 +3520,7 @@ is currently ignored.
</table>
<<page foot>>
@
+
\subsection{aldorusersguidepage.xhtml}
<<aldorusersguidepage.xhtml>>=
<<standard head>>
@@ -5272,6 +5742,31 @@ the first k 20th powers.
<<page foot>>
@
+\subsection{cats.xhtml}
+<<cats.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ CATS -- Computer Algebra Test Suite
+ </div>
+<hr/>
+The Computer Algebra Test Suite is intended to show that Axiom conforms
+to various published standards. Axiom implementations of these functions
+are tested against reference publications.
+
+In order to show standards compliance we need to examine Axiom's behavior
+against known good results. Where possible, these results are also tested
+against other available computer algebra systems.
+
+The available test suites are:
+<ol>
+ <li><a href="dlmf.xhtml">Gamma Function</a></li>
+</ol>
+<<page foot>>
+@
+
\subsection{commandline.xhtml}
<<commandline.xhtml>>=
<<standard head>>
@@ -9256,6 +9751,20361 @@ the operations will have extra ones added at some
stage.
<<page foot>>
@
+\subsection{dlmf.xhtml}
+<<dlmf.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function by R. A. Askey and R. Roy
+ </div>
+ <hr/>
+<p>
+The Gamma function is an extension of the factorial function to
+real and complex numbers. For positive integers,
+<m:math display="inline">
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>n</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+</m:math>.
+</p>
+
+<p>
+These pages explore Axiom's facilities for handling the Gamma function.
+In particular we try to show that Axiom conforms to published standards.
+</p>
+<ul>
+ <li><b>Notation</b></li>
+ <ul>
+ <li><a href="dlmfnotation.xhtml">Notation</a></li>
+ </ul>
+ <li><b>Properties</b></li>
+ <ul>
+ <li><a href="dlmfdefinitions.xhtml">Definitions</a></li>
+ <li><a href="dlmfgraphics.xhtml">Graphics</a></li>
+ <li><a href="dlmfspecialvaluesandextrema.xhtml">
+ Special Values and Extrema</a></li>
+ <li><a href="dlmffunctionrelations.xhtml">Function Relations</a></li>
+ <li><a href="dlmfinequalities.xhtml">Inequalities</a></li>
+ <li><a href="dlmfseriesexpansions.xhtml">Series Expansions</a></li>
+ <li><a href="dlmfinfiniteproducts.xhtml">Infinite Products</a></li>
+ <li><a href="dlmfintegralrepresentations.xhtml">
+ Integral Representations</a></li>
+ <li><a href="dlmfcontinuedfractions.xhtml">Continued Fractions</a></li>
+ <li><a href="dlmfasymptoticexpansions.xhtml">Asymptotic Expansions</a></li>
+ <li><a href="dlmfbetafunction.xhtml">Beta Function</a></li>
+ <li><a href="dlmfintegrals.xhtml">Integrals</a></li>
+ <li><a href="dlmfmultidimensionalintegral.xhtml">
+ Multidimensional Integral</a></li>
+ <li><a href="dlmfpolygammafunctions.xhtml">Polygamma Functions</a></li>
+ <li><a href="dlmfsums.xhtml">Sums</a></li>
+ <li><a href="dlmfbarnesgfunction.xhtml">
+ Barnes <i>G</i>-Function (Double Gamma Function)</a></li>
+ <li><a href="dlmfqgammaandbetafunctions.xhtml">
+ <i>q</i>-Gamma and Beta Functions</a></li>
+ </ul>
+ <li><b>Applications</b></li>
+ <ul>
+ <li><a href="dlmfmathematicalapplications.xhtml">
+ Mathematical Applications</a></li>
+ <li><a href="dlmfphysicalapplications.xhtml">
+ Physical Applications</a></li>
+ </ul>
+ <li><b>Computation</b></li>
+ <ul>
+ <li><a href="dlmfmethodsofcomputation.xhtml">
+ Methods of Computation</a></li>
+ <li><a href="dlmftables.xhtml">Tables</a></li>
+ <li><a href="dlmfapproximations.xhtml">Approximations</a></li>
+ <li>Axiom Software</li>
+ </ul>
+</ul>
+<<page foot>>
+@
+
+\subsection{dlmfapproximations.xhtml}
+<<dlmfapproximations.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Approximations
+ </div>
+ <hr/>
+<h3>Approximations</h3>
+<h6>Contents</h6>
+<ul>
+ <li>Rational Approximations</li>
+ <li>Expansions in Chebyshev Series</li>
+ <li>Approximations in the Complex Plane</li>
+</ul>
+
+<h4>Rational Approximations</h4>
+
+<p>
+ <a href="http://dlmf.nist.gov/Contents/bib/C#cody:1967:ca";>
+ Cody and Hillstrom(1967)
+ </a> gives minimax rational approximations for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math> for the ranges
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0.5</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>1.5</m:mn>
+ </m:mrow>
+ </m:math>,
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>1.5</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>4</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>4</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>12</m:mn>
+ </m:mrow>
+ </m:math>; precision is variable.
+ <a href="http://dlmf.nist.gov/Contents/bib/H#hart:1968:ca";>
+ Hart <em>et.al.</em>(1968)
+ </a> gives minimax polynomial and rational approximations to
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math> in the intervals
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>8</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>1000</m:mn>
+ </m:mrow>
+ </m:math>,
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>12</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>1000</m:mn>
+ </m:mrow>
+ </m:math>; precision is variable.
+
+ <a href="http://dlmf.nist.gov/Contents/bib/C#cody:1973:cap";>
+ Cody <em>et.al.</em>(1973)
+ </a> gives minimax rational approximations for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> for the ranges
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0.5</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ </m:math> and
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>3</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo><</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math>; precision is variable.
+</p>
+
+<p>For additional approximations see
+ <a href="http://dlmf.nist.gov/Contents/bib/H#hart:1968:ca";>
+ Hart <em>et.al.</em>(1968)
+ </a>(Appendix B),
+ <a href="http://dlmf.nist.gov/Contents/bib/L#luke:1975:mfa";>
+ Luke(1975)
+ </a>(pp. 22–23), and
+ <a href="http://dlmf.nist.gov/Contents/bib/W#weniger:2003:dig";>
+ Weniger(2003)
+ </a>.
+</p>
+
+<h4>Expansions in Chebyshev Series</h4>
+
+<p>
+ <a href="http://dlmf.nist.gov/Contents/bib/L#luke:1969:sfa2";>
+ Luke(1969)
+ </a>
+ gives the coefficients to 20D for the Chebyshev-series expansions of
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>x</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>x</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:math>,
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>, and the first six derivatives of
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> for
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>. These coefficients are reproduced in
+ <a href="http://dlmf.nist.gov/Contents/bib/L#luke:1975:mfa";>
+ Luke(1975)
+ </a>.
+
+ <a href="http://dlmf.nist.gov/Contents/bib/C#clenshaw:1962:csm";>
+ Clenshaw(1962)
+ </a> also gives 20D Chebyshev-series coefficients for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>x</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> and its reciprocal for
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>. See
+ <a href="http://dlmf.nist.gov/Contents/bib/L#luke:1975:mfa";>
+ Luke(1975)
+ </a>(pp. 22–23) for additional expansions.
+</p>
+
+<h4>Approximations in the Complex Plane</h4>
+
+<p>Rational approximations for
+ <m:math display="inline">
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>A</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:math>, where
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>A</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>, and approximations for
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> based on the Padé approximants for two forms of the incomplete
+ gamma function are in
+ <a href="http://dlmf.nist.gov/Contents/bib/L#luke:1969:sfa2";>
+ Luke(1969)
+ </a>.
+ <a href="http://dlmf.nist.gov/Contents/bib/L#luke:1975:mfa";>
+ Luke(1975)
+ </a>(pp. 13–16) provides explicit rational approximations for
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ </m:math>
+</p>
+<<page foot>>
+@
+
+\subsection{dlmfasymptoticexpansions.xhtml}
+<<dlmfasymptoticexpansions.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Asymptotic Expansions
+ </div>
+ <hr/>
+<h3>Asymptotic Expansions</h3>
+
+<h6>Contents</h6>
+<ul>
+ <li>Poincaré-Type Expansions</li>
+ <li>Error Bounds and Exponential Improvement</li>
+ <li>Ratios</li>
+</ul>
+
+<h4>Poincaré-Type Expansions</h4>
+
+<p>As
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math> in the sector
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>δ</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<a name="equation1"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>B</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msub>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<a name="equation2"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>B</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msub>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>For the Bernoulli numbers
+ <m:math>
+ <m:msub>
+ <m:mi>B</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msub>
+ </m:math>,
+ Also,
+</p>
+
+<a name="equation3"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>z</m:mi>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mi>z</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>288</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>3</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>139</m:mn>
+ <m:mn>51840</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>4</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>571</m:mn>
+ <m:mn>24 88320</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>5</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>1 63879</m:mn>
+ <m:mn>2090 18880</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>6</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>52 46819</m:mn>
+ <m:mn>7 52467 96800</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msqrt>
+ <m:mn>2</m:mn>
+ </m:msqrt>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:msqrt>
+ <m:mn>2</m:mn>
+ </m:msqrt>
+ </m:mrow>
+ </m:mrow>
+ </m:math>, and
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>k</m:mi>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>≥</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<p>
+ <a href="http://dlmf.nist.gov/Contents/bib/W#wrench:1968:cts";>
+ Wrench(1968)
+ </a> gives exact values of
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:math> up to
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mn>20</m:mn>
+ </m:msub>
+ </m:math>.
+ <a href="http://dlmf.nist.gov/Contents/bib/S#spira:1971:cot";>
+ Spira(1971)
+ </a>
+ corrects errors in Wrench's results and also supplies exact and 45D values of
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:math> for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>21</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>22</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:mn>30</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math>. For an asymptotic expansion of
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:math> as
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math> see
+ <a href="http://dlmf.nist.gov/Contents/bib/B#boyd:1994:gfa";>Boyd(1994)
+ </a>.
+</p>
+
+<p>With the same conditions
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:msqrt>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:msqrt>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo>∈</m:mo>
+ <m:mi mathvariant="normal">ℂ</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> are both fixed, and
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>h</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>h</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:msub>
+ <m:mi>B</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>h</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>h</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo>∈</m:mo>
+ <m:mrow>
+ <m:mo>[</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>]</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> is fixed.
+</p>
+
+<p>Also as
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>y</m:mi>
+ <m:mo>→</m:mo>
+ <m:mrow>
+ <m:mo>±</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:msqrt>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:msqrt>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>y</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>y</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>uniformly for bounded real values of
+ <m:math display="inline">
+ <m:mi>x</m:mi>
+ </m:math>.
+</p>
+
+<h4>Error Bounds and Exponential Improvement</h4>
+
+<p>If the sums in the expansions
+(<a href="#equation1">Equation 1</a>) and
+(<a href="#equation2">Equation 2</a>) are terminated at
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math> (
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>≥</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>) and
+ <m:math display="inline">
+ <m:mi>z</m:mi>
+ </m:math>
+is real and positive, then the remainder terms are bounded in magnitude by
+the first neglected terms and have the same sign. If
+ <m:math display="inline">
+ <m:mi>z</m:mi>
+ </m:math>
+is complex, then the remainder terms are bounded in magnitude by
+ <m:math display="inline">
+ <m:mrow>
+ <m:msup>
+ <m:mi>sec</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> for
+(<a href="#equation1">Equation 1</a>), and
+ <m:math display="inline">
+ <m:mrow>
+ <m:msup>
+ <m:mi>sec</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> for
+(<a href="#equation2">Equation 2</a>), times the first neglected terms.</p>
+
+<p>For the remainder term in
+(<a href="#equation3">Equation 3</a>) write
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>z</m:mi>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mi>z</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>K</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:munderover>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>g</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>R</m:mi>
+ <m:mi>K</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>K</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<p>Then
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>R</m:mi>
+ <m:mi>K</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>K</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>K</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>K</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mi>K</m:mi>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mo movablelimits="false">min</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>sec</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:msup>
+ <m:mi>K</m:mi>
+ <m:mstyle scriptlevel="+1">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+
+<h4>Ratios</h4>
+
+<p>If
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo>∈</m:mo>
+ <m:mi mathvariant="normal">ℂ</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo>∈</m:mo>
+ <m:mi mathvariant="normal">ℂ</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> are fixed as
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math> in
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>δ</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>, then
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>∼</m:mo>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mi>G</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>Also, with the added condition
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mi>H</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>Here
+</p>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>G</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>G</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>G</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="true">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="true">
+ <m:mtable rowspacing="0.2ex" columnspacing="0.4em">
+ <m:mtr>
+ <m:mtd>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mtd>
+ </m:mtr>
+ <m:mtr>
+ <m:mtd>
+ <m:mn>2</m:mn>
+ </m:mtd>
+ </m:mtr>
+ </m:mtable>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>3</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>H</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>H</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="true">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="true">
+ <m:mtable rowspacing="0.2ex" columnspacing="0.4em">
+ <m:mtr>
+ <m:mtd>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mtd>
+ </m:mtr>
+ <m:mtr>
+ <m:mtd>
+ <m:mn>2</m:mn>
+ </m:mtd>
+ </m:mtr>
+ </m:mtable>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>H</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="true">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>240</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="true">
+ <m:mtable rowspacing="0.2ex" columnspacing="0.4em">
+ <m:mtr>
+ <m:mtd>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mtd>
+ </m:mtr>
+ <m:mtr>
+ <m:mtd>
+ <m:mn>4</m:mn>
+ </m:mtd>
+ </m:mtr>
+ </m:mtable>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mn>5</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>In terms of generalized Bernoulli polynomials we have for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>G</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mtable rowspacing="0.2ex" columnspacing="0.4em">
+ <m:mtr>
+ <m:mtd>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mtd>
+ </m:mtr>
+ <m:mtr>
+ <m:mtd>
+ <m:mi>k</m:mi>
+ </m:mtd>
+ </m:mtr>
+ </m:mtable>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mi>B</m:mi>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>H</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mtable rowspacing="0.2ex" columnspacing="0.4em">
+ <m:mtr>
+ <m:mtd>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mtd>
+ </m:mtr>
+ <m:mtr>
+ <m:mtd>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:mtd>
+ </m:mtr>
+ </m:mtable>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mi>B</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfbarnesgfunction.xhtml}
+<<dlmfbarnesgfunction.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Barnes G-Function (Double Gamma Function)
+ </div>
+ <hr/>
+<h3>Barnes
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">G</m:mi>
+ </m:math>-Function (Double Gamma Function)
+</h3>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>G</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>G</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>G</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>G</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>n</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>G</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>γ</m:mi>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>×</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>Ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>G</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>Ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi>z</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi>Ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>t</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>The
+ <m:math display="inline">
+ <m:mi>Ln</m:mi>
+ </m:math>'s have their principal values on the positive real axis and are
+ continued via continuity.
+</p>
+
+<p>When
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math> in
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>δ</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>Ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>G</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>4</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>Ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mi>A</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>B</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:msub>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>see
+<a href="http://dlmf.nist.gov/Contents/bib/F#ferreira:2001:aae";>
+ Ferreira and López(2001)
+</a>. This reference also provides bounds for the error term. Here
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>B</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:math> is the Bernoulli number, and
+ <m:math display="inline">
+ <m:mi>A</m:mi>
+ </m:math> is <em>Glaisher's constant</em>, given by
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>A</m:mi>
+ <m:mo>=</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mi>C</m:mi>
+ </m:msup>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1.28242 71291 00622 63687</m:mn>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>C</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munder>
+ <m:mo movablelimits="false">lim</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:munder>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:mi>k</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msup>
+ <m:mi>n</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>4</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msup>
+ <m:mi>n</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>γ</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ζ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:msup>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ζ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>and
+ <m:math display="inline">
+ <m:msup>
+ <m:mi>ζ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ </m:math> is the derivative of the zeta function
+</p>
+
+<p>For Glaisher's constant see also
+ <a href="http://dlmf.nist.gov/Contents/bib/G#greene:1982:mft";>
+ Greene and Knuth(1982)
+ </a>(p. 100).
+</p>
+<<page foot>>
+@
+
+\subsection{dlmfbetafunction.xhtml}
+<<dlmfbetafunction.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Beta Function
+ </div>
+ <hr/>
+<h3>Beta Function</h3>
+
+<p>In this section all fractional powers have their principal values, except
+where noted otherwise. In the next 4 equations it is assumed
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math> and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>.
+</p>
+
+<h5>Euler's Beta Integral</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mn>1</m:mn>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mfrac bevelled="true">
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msubsup>
+ <m:mrow>
+ <m:msup>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mi>θ</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>cos</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mi>θ</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>θ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mn>1</m:mn>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>t</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>with
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:math> and the integration path along the real axis.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mfrac bevelled="true">
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msubsup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:msup>
+ <m:mn>2</m:mn>
+ <m:mi>a</m:mi>
+ </m:msup>
+ </m:mfrac>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi>π</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>b</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:msup>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ <m:mfrac>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>π</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>cosh</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>b</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>cosh</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mn>4</m:mn>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>w</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>a</m:mi>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>b</m:mi>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>w</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>w</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>The fractional powers have their principal values when
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>w</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math> and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>, and are continued via continuity.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>t</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mi>π</m:mi>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <!-- Need a better Axiom graphic for this
+ <img width="302" height="151" alt="" src="bitmaps/12F1.png"/> -->
+</div>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mi>t</m:mi>
+ </m:math>-plane. Contour for first loop integral for the beta function.
+</div>
+
+<p>In the next two equations the fractional powers are continuous on the
+ integration paths and take their principal values at the beginning.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p> when
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math>
+ <m:mi>a</m:mi>
+ </m:math> is not an integer and the contour cuts the real axis between
+ <m:math>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math> and the origin.
+</p>
+
+<div align="center">
+ <!-- Need a better Axiom graphic for this
+ <img width="302" height="151" alt="" src="bitmaps/12F2.png"/> -->
+</div>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mi>t</m:mi>
+ </m:math>-plane. Contour for second loop integral for the beta function.
+</div>
+
+<h5>Pochhammer's Integral</h5>
+<p>When
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>∈</m:mo>
+ <m:mi mathvariant="normal">ℂ</m:mi>
+ </m:mrow>
+ </m:math>
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mi>P</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>-</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>4</m:mn>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where the contour starts from an arbitrary point
+ <m:math display="inline">
+ <m:mi>P</m:mi>
+ </m:math> in the interval
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:math>,circles
+ <m:math display="inline">
+ <m:mn>1</m:mn>
+ </m:math> and then
+ <m:math display="inline">
+ <m:mn>0</m:mn>
+ </m:math> in the positive sense, circles
+ <m:math display="inline">
+ <m:mn>1</m:mn>
+ </m:math> and then
+ <m:math display="inline">
+ <m:mn>0</m:mn>
+ </m:math> in the negative sense, and returns to
+ <m:math display="inline">
+ <m:mi>P</m:mi>
+ </m:math>. It can always be deformed into the contour shown here.
+</p>
+
+<div align="center">
+ <!-- Need a better Axiom graphic for this
+ <img width="302" height="104" alt="" src="bitmaps/12F3.png"/> -->
+</div>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mi>t</m:mi>
+ </m:math>-plane. Contour for Pochhammer's integral.
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfcontinuedfractions.xhtml}
+<<dlmfcontinuedfractions.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Continued Fractions
+ </div>
+ <hr/>
+<h3>Continued Fractions</h3>
+
+<p>For
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>3</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>4</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>5</m:mn>
+ </m:msub>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>12</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>30</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>53</m:mn>
+ <m:mn>210</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>3</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>195</m:mn>
+ <m:mn>371</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>4</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>22999</m:mn>
+ <m:mn>22737</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>5</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>299 44523</m:mn>
+ <m:mn>197 33142</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>6</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mn>10 95352 41009</m:mn>
+ <m:mn>4 82642 75462</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>For rational values of
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>7</m:mn>
+ </m:msub>
+ </m:math> to
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>11</m:mn>
+ </m:msub>
+ </m:math> and 40S values of
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>0</m:mn>
+ </m:msub>
+ </m:math> to
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>40</m:mn>
+ </m:msub>
+ </m:math>, see
+<a href="http://dlmf.nist.gov/Contents/bib/C#char:1980:osc";>
+ Char(1980)
+</a>. Also see
+<a href="http://dlmf.nist.gov/Contents/bib/J#jones:1980:con";>
+ Jones and Thron(1980)
+</a>(pp. 348–350) and
+<a href="http://dlmf.nist.gov/Contents/bib/L#lorentzen:1992:cfa";>
+ Lorentzen and Waadeland(1992)
+</a>(pp. 221–224) for further information.
+</p>
+<<page foot>>
+@
+
+\subsection{dlmfdefinitions.xhtml}
+<<dlmfdefinitions.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Definitions
+ </div>
+ <hr/>
+<h3>Definitions</h3>
+<h6>Contents</h6>
+<ul>
+ <li>Gamma and Psi Functions</li>
+ <li>Euler's Constant</li>
+ <li>Pochhammer's Symbol</li>
+</ul>
+<h4>Gamma and Psi Functions</h4>
+<h5>Euler's Integral</h5>
+<m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+</m:math>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+
+When
+<m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+</m:math>,
+
+<m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+</m:math> is defined by analytic continuation. It is a meromorphic
+ function with no zeros, and with simple poles of residue
+
+<m:math display="inline">
+ <m:mfrac bevelled="true">
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:mfrac>
+</m:math> at
+
+<m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+</m:math>.
+
+<m:math display="inline">
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+</m:math> is entire, with simple zeros at
+
+<m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+</m:math>.
+
+<p>
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+</p>
+
+<p>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> is meromorphic with simple poles of residue
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math> at
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</p>
+
+<h4>Euler's Constant</h4>
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>γ</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munder>
+ <m:mo movablelimits="false">lim</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:munder>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>0.57721 56649 01532 86060</m:mn>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+
+<h4>Pochhammer's Symbol</h4>
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>0</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ </div>
+<<page foot>>
+@
+
+\subsection{dlmffunctionrelations.xhtml}
+<<dlmffunctionrelations.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Function Relations
+ </div>
+ <hr/>
+<h3>Functional Relations</h3>
+<h6>Contents</h6>
+<ul>
+ <li>Recurrence</li>
+ <li>Reflection</li>
+ <li>Multiplication</li>
+ <li>Bohr-Mollerup Theorem</li>
+</ul>
+<h4>Recurrence</h4>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>z</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+
+<h4>Reflection</h4>
+<a name="equation3"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>ŷ</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mi>tan</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>ŷ</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<h4>Multiplication</h4>
+<div align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mn>3</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>3</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mn>3</m:mn>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>3</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>2</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:msup>
+ <m:mi>n</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:munderover>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mi>k</m:mi>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:munderover>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>k</m:mi>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:msup>
+ <m:mi>n</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:munderover>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mi>k</m:mi>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<a name="bohrmolleruptheorem"/>
+<h4>Bohr-Mollerup Theorem</h4>
+
+<br/>
+If a positive function
+<m:math display="inline">
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+</m:math> on
+<m:math display="inline">
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+</m:math> satisfies
+<m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+</m:math>,
+
+<m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+</m:math>, and
+
+<m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+</m:math> is convex, then
+
+<m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+</m:math>.
+<<page foot>>
+@
+
+\subsection{dlmfgraphics.xhtml}
+<<dlmfgraphics.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Graphics
+ </div>
+ <hr/>
+<h3>Graphics</h3>
+<h6>Contents</h6>
+<ul>
+ <li>Real Argument</li>
+ <li>The Psi Function</li>
+ <li>Complex Argument</li>
+</ul>
+<h4>Real Argument</h4>
+ <img width="403" height="482" src="bitmaps/gammareal3.png"/>
+ <br/>
+This graph shows the
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> and
+ <m:math display="inline">
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:math>.
+
+To create these two graphs in Axiom:
+<pre>
+ -- Draw the first graph in a viewport
+ viewport1:=draw(Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+ -- Draw the second graph in a viewport
+ viewport2:=draw(1/Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+ -- Get the Gamma graph from the first viewport and layer it on top
+ putGraph(viewport2,getGraph(viewport1,1),2)
+ -- Remove the points and leave the lines
+ points(viewport2,1,"off")
+ points(viewport2,2,"off")
+ -- Show the combined graph
+ makeViewport2D(viewport2)
+</pre>
+
+ <img width="300" height="176" alt="" src="bitmaps/loggamma.png"/>
+ <br/>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>. This function is convex on
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:math>;
+ <br/>
+ compare <a href="dlmffunctionalrelations.xhtml#bohrmolleruptheorem">
+ Functional Relations</a>
+ <p>
+You can construct this graph with the Axiom commands:
+<pre>
+ -- draw the graph of log(Gamma) in a viewport
+ viewport1:=draw(log Gamma(i), i=0..8, adaptive==true, unit==[1.0,1.0])
+ -- turn off the points and leave the lines
+ points(viewport1,1,"off")
+</pre>
+</p>
+ <br/>
+
+ <h4>The Psi Function
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</h4>
+
+<p> This function is a special case of the polygamma function.
+In particular,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> is equal to polygamma(0,x).
+ </p>
+ <br/>
+ <br/>
+
+ <img width="522" height="556" alt="" src="bitmaps/psi.png"/>
+ <br/>
+You can reconstruct this graph in Axiom by:
+<pre>
+ -- first construct the psi function
+ psi(x)==polygamma(0,x)
+ -- draw the graph in a viewport
+ viewport:=draw(psi(y),y=-3.5..4,adaptive==true)
+ -- make the gradient obvious
+ scale(viewport,1,0.9,22.5)
+ -- and recenter the graph
+ translate(viewport,1,0,-0.02)
+ -- turn off the points and keep the line
+ points(viewport,1,"off")
+</pre>
+
+ <h4>Complex Argument</h4>
+
+ <img width="400" height="400" alt="" src="bitmaps/gammacomplex.png"/>
+ <br/>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+ <br/>
+
+You can reconstruct this image in Axiom with:
+<pre>
+ -- Set up the default viewpoint
+ viewPhiDefault(-%pi/4)
+ -- define the point set function
+ gam(x,y)==
+ g:=Gamma complex(x,y)
+ point [x,y,max(min(real g,4),-4), argument g]
+ -- draw the image and remember the viewport
+ viewport:=draw(gam, -4..4,-3..3,var1Steps==100,var2Steps==100)
+ -- set the color mapping for the image
+ colorDef(viewport,blue(),blue())
+ -- and smoothly shade it
+ drawStyle(viewport,"smooth")
+</pre>
+ <img width="400" height="400" src="bitmaps/gammacomplexinverse.png"/>
+<br/>
+ <m:math display="inline">
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:math>
+ <br/>
+
+<p>
+You can reproduce this image from Axiom with:
+<pre>
+ -- Set up the default viewpoint
+ viewPhiDefault(-%pi/4)
+ -- Define the complex Gamma inverse function
+ gaminv(x,y)==
+ g:=1/(Gamma complex(x,y))
+ point [x,y,max(min(real g,4),-4), argument g]
+ -- draw the 3D image and remember the viewport
+ viewport:=draw(gaminv, -4..4,-3..3,var1Steps==100,var2Steps==100)
+ -- make the image a uniform color
+ colorDef(viewport,blue(),blue())
+ -- and make it pretty
+ drawStyle(viewport,"smooth")
+</pre>
+</p>
+
+
+<p>
+To get these exact images with the colored background you need
+to use GIMP to set the background. The steps I used are:
+<ol>
+<li>Save the image as a pixmap</li>
+<li>Open the saved file in gimp</li>
+<li>Dialogs->Colors->ColorPicker button</li>
+<li>Eyedrop the color of the web page</li>
+<li>Set the color as the foreground on the FG/BG page</li>
+<li>Dialogs->Layers</li>
+<li>Duplicate Layer</li>
+<li>Layer->Stack->Select bottom layer</li>
+<li>Edit->Fill with Foreground color</li>
+<li>(on Layers panel)Select image</li>
+<li>(on Layers panel) Mode->Darken Only</li>
+</ol>
+Note that you may have to use "lighten only" first before it will
+allow you to choose "darken only".
+</p>
+
+<<page foot>>
+@
+
+\subsection{dlmfinequalities.xhtml}
+<<dlmfinequalities.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Inequalities
+ </div>
+ <hr/>
+<h3>Inequalities</h3>
+<h6>Contents</h6>
+<ul>
+ <li>Real Variables</li>
+ <li>Complex Variables</li>
+</ul>
+
+<h4>Real Variables</h4>
+<p>Throughout this subsection
+<m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+</m:math>.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo><</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi>x</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>x</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mi>x</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>12</m:mn>
+ <m:mi>x</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mi>x</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mi>x</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>x</m:mi>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mo><</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo><</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>s</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:msup>
+ <m:mi>s</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>s</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>s</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h4>Complex Variables</h4>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≥</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>sech</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>≥</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>For
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>≥</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>≥</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>, and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math> with
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>∣</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>∣</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>For
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>≥</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>y</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>6</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfinfiniteproducts.xhtml}
+<<dlmfinfiniteproducts.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Infinite Products
+ </div>
+ <hr/>
+<h3>Infinite Products</h3>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munder>
+ <m:mo movablelimits="false">lim</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:munder>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:msup>
+ <m:mi>k</m:mi>
+ <m:mi>z</m:mi>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mi>γ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:msup>
+ <m:mi>y</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>m</m:mi>
+ </m:munderover>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>m</m:mi>
+ </m:munderover>
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>then
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>m</m:mi>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mi>m</m:mi>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mi>m</m:mi>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>m</m:mi>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>provided that none of the
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>b</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:math>
+ is zero or a negative integer.
+</p>
+<<page foot>>
+@
+
+\subsection{dlmfintegrals.xhtml}
+<<dlmfintegrals.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Integrals
+ </div>
+ <hr/>
+<h3>Integrals</h3>
+
+<a name="equation1"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>s</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>a</m:mi>
+ </m:msup>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo><</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<h5>Barnes's Beta Integral</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>d</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h5>Ramanujan's Beta Integral</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>d</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>d</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<h5>de Branges-Wilson Beta Integral</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>4</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo></m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mn>4</m:mn>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∏</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>j</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mn>4</m:mn>
+ </m:mrow>
+ </m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>j</m:mi>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>3</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mn>4</m:mn>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:mn>4</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfintegralrepresentations.xhtml}
+<<dlmfintegralrepresentations.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Integral Representations
+ </div>
+ <hr/>
+<h3>Integral Representations</h3>
+
+<h6>Contents</h6>
+<ul>
+ <li>Gamma Function</li>
+ <li>Psi Function and Euler's Constant</li>
+</ul>
+
+<h4>Gamma Function</h4>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>μ</m:mi>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>ν</m:mi>
+ <m:mi>μ</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mi>ν</m:mi>
+ <m:mi>μ</m:mi>
+ </m:mfrac>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mi>μ</m:mi>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>ν</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>ν</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>μ</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>, and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>. (The fractional powers have their principal values.)
+</p>
+
+<h5>Hankel's Loop Integral</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>+</m:mo>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msubsup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mi>t</m:mi>
+ </m:msup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where the contour begins at
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math>, circles the origin once in the positive direction, and returns to
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math>.
+ <m:math display="inline">
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:math> has its principal value where
+ <m:math display="inline">
+ <m:mi>t</m:mi>
+ </m:math> crosses the positive real axis, and is continuous.
+</p>
+
+<div align="center">
+ <!-- need a better Axiom graphic than this
+ <img width="302" height="150" alt="" src="bitmaps/9F1.png"/> -->
+</div>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mi>t</m:mi>
+ </m:math>-plane. Contour for Hankel's loop integral.
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>c</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>t</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>c</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where the path is the real axis.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>1</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>π</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>π</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msup>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>4</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msup>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>4</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<h5>Binet's Formula</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>arctan</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mi>t</m:mi>
+ <m:mi>z</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mo>⁡</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:math> and the inverse tangent has its principal value.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>γ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>s</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mo>⁡</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>δ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math> (
+ <m:math display="inline">
+ <m:mrow>
+ <m:none/>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ </m:math>),
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:math>, and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>s</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</p>
+
+<p>For additional representations see
+<a href="http://dlmf.nist.gov/Contents/bib/W#whittaker:1927:cma";>
+ Whittaker and Watson(1927)</a>
+</p>
+
+<h4>Psi Function and Euler's Constant</h4>
+<p>For
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mi>t</m:mi>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>t</m:mi>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>t</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>z</m:mi>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mi>t</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:mo>+</m:mo>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mn>1</m:mn>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>sin</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>s</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>δ</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math> and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>c</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:math>.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>γ</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mi>t</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mn>1</m:mn>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mi>t</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>1</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ <m:mi>t</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mi>t</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfmathematicalapplications.xhtml}
+<<dlmfmathematicalapplications.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Mathematical Applications
+ </div>
+ <hr/>
+<h3>Mathematical Applications</h3>
+<h6>Contents</h6>
+<ul>
+ <li>Summation of Rational Functions</li>
+ <li>Mellin-Barnes Integrals</li>
+ <li>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">n</m:mi>
+ </m:math>-Dimensional Sphere</li>
+</ul>
+
+<h4>Summation of Rational Functions</h4>
+
+<p>As shown in
+ <a href="http://dlmf.nist.gov/Contents/bib/T#temme:1996:sfi";>
+ Temme(1996)
+ </a>(§3.4), the results given in
+ <a href="dlmfseriesexpansions.xhtml">
+ Series Expansions
+ </a> can be used to sum infinite series of rational functions.
+</p>
+
+<h5>Example</h5>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>S</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ </m:mstyle>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mstyle displaystyle="true">
+ <m:mfrac>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>3</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>By decomposition into partial fractions</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>a</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>2</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>2</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>Hence from (
+ <a href="dlmfseriesexpansions.xhtml#equation6">Series Expansions 6
+ </a>), ( Special Values and Extrema
+ <a href="dlmfspecialvaluesandextrema.xhtml#equation13">
+ Equation 13
+ </a> and
+ <a href="dlmfspecialvaluesandextrema.xhtml#equation19">
+ Equation 19
+ </a>)
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>S</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>2</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>3</m:mn>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>π</m:mi>
+ <m:msqrt>
+ <m:mn>3</m:mn>
+ </m:msqrt>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h4>Mellin-Barnes Integrals</h4>
+<p>Many special functions
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> can be represented as a <em>Mellin-Barnes integral</em>, that is,
+ an integral of a product of gamma functions, reciprocals of gamma
+ functions, and a power of
+ <m:math display="inline">
+ <m:mi>z</m:mi>
+ </m:math>, the integration contour being doubly-infinite and eventually
+ parallel to the imaginary axis. The left-hand side of (
+ <a href="dlmfintegrals.xhtml#equation1">
+ Integral Equation 1
+ </a>) is a typical example. By translating the contour parallel to itself
+ and summing the residues of the integrand, asymptotic expansions of
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>f</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> for large
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:math>, or small
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:math>, can be obtained complete with an integral representation of the
+ error term.
+</p>
+
+<h4>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">n</m:mi>
+ </m:math>-Dimensional Sphere</h4>
+
+<p>The volume
+ <m:math display="inline">
+ <m:mi>V</m:mi>
+ </m:math> and surface area
+ <m:math display="inline">
+ <m:mi>A</m:mi>
+ </m:math> of the
+ <m:math display="inline">
+ <m:mi>n</m:mi>
+ </m:math>-dimensional sphere of radius
+ <m:math display="inline">
+ <m:mi>r</m:mi>
+ </m:math> are given by
+</p>
+
+<div align="center">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>V</m:mi>
+ <m:mo>=</m:mo>
+ <m:mstyle displaystyle="true">
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi>r</m:mi>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>S</m:mi>
+ <m:mo>=</m:mo>
+ <m:mstyle displaystyle="true">
+ <m:mfrac>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:msup>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:msup>
+ <m:mi>r</m:mi>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="true">
+ <m:mfrac>
+ <m:mi>n</m:mi>
+ <m:mi>r</m:mi>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>V</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfmethodsofcomputation.xhtml}
+<<dlmfmethodsofcomputation.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Methods of Computation
+ </div>
+ <hr/>
+<h3>Methods of Computation</h3>
+
+<p>An effective way of computing
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+in the right half-plane is backward recurrence, beginning with a value
+generated from the
+<a href="dlmfasymptoticexpansions.xhtml#equation3">
+ asymptotic expansion
+</a>
+Or we can use forward recurrence, with an
+<a href="dlmfseriesexpansions.xhtml#equation3">
+ initial value
+</a>.
+For the left half-plane we can continue the backward recurrence or
+make use of the
+<a href="dlmffunctionrelations.xhtml#equation3">
+ reflection formula
+</a>.
+</p>
+
+<p>Similarly for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>, and the polygamma functions.
+</p>
+
+<p>For a comprehensive survey see
+ <a href="http://dlmf.nist.gov/Contents/bib/V#vanderlaan:1984:csf";>
+ van der Laan and Temme(1984)
+ </a>(Chapter III).
+ See also
+ <a href="http://dlmf.nist.gov/Contents/bib/B#borwein:1992:feg";>
+ Borwein and Zucker(1992)
+ </a>.
+</p>
+<<page foot>>
+@
+
+\subsection{dlmfmultidimensionalintegral.xhtml}
+<<dlmfmultidimensionalintegral.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Multidimensional Integral
+ </div>
+ <hr/>
+<h3>Multidimensional Integrals</h3>
+
+<p>Let
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>V</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:math> be the simplex:
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>≥</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>. Then for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msub>
+ <m:mi>V</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:msub>
+ <m:msubsup>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msubsup>
+ <m:msubsup>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msubsup>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:msubsup>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ <m:mrow>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msub>
+ <m:mi>V</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:msub>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:msubsup>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>+</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>+</m:mo>
+ <m:msub>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h5>Selberg-type Integrals</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Δ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munder>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>j</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:munder>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>j</m:mi>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>Then
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>[</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>]</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:msub>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>m</m:mi>
+ </m:msub>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Δ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:msubsup>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msubsup>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:mfrac>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>m</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>provided that
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>min</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mi>b</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</p>
+
+<p>Secondly,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>[</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:msub>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>m</m:mi>
+ </m:msub>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Δ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:msubsup>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msubsup>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>m</m:mi>
+ </m:munderover>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msubsup>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mo></m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>when
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>min</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mi>a</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</p>
+
+<p>Thirdly,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mi>n</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ </m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:msub>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Δ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msubsup>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ <m:mn>2</m:mn>
+ </m:msubsup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>t</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msubsup>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>c</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h5>Dyson's Integral</h5>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>[</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>]</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:msub>
+ <m:munder>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi>j</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:munder>
+ <m:msup>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mi>j</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>b</m:mi>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mi>n</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfnotation.xhtml}
+<<dlmfnotation.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">Digital Library of Mathematical Functions<br/>
+ The Gamma Function -- Notation
+ </div>
+ <hr/>
+ <div class="content">
+ <div class="section">
+ <h3>Notation</h3>
+ <div class="table" id="T1">
+ <table align="center">
+ <tbody>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>j</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>m</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="justify">nonnegative integers.</td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mi>k</m:mi>
+ </m:math>
+ </th>
+ <td>except in <a href="dlmfphysicalapplications.xhtml">
+ Physical Applications</a>
+ </td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="justify">real variables.</td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="justify">complex variable.</td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>q</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>s</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>w</m:mi>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="justify">real or complex variables with
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>∣</m:mo>
+ <m:mi>q</m:mi>
+ <m:mo>∣</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>.
+ </td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mi>δ</m:mi>
+ </m:math>
+ </th>
+ <td align="justify">arbitrary small positive constant.</td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mi mathvariant="normal">ℂ</m:mi>
+ </m:math>
+ </th>
+ <td align="justify">complex plane (excluding infinity).</td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mi mathvariant="normal">ℝ</m:mi>
+ </m:math>
+ </th>
+ <td align="justify">real line (excluding infinity).</td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle scriptlevel="+1">
+ <m:mtable rowspacing="0.2ex" columnspacing="0.4em">
+ <m:mtr>
+ <m:mtd>
+ <m:mi>n</m:mi>
+ </m:mtd>
+ </m:mtr>
+ <m:mtr>
+ <m:mtd>
+ <m:mi>m</m:mi>
+ </m:mtd>
+ </m:mtr>
+ </m:mtable>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="justify">binomial coefficient
+ <m:math display="inline">
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>m</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>m</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:math>.
+ </td>
+ </tr>
+ <tr>
+ <th align="left">empty sums</th>
+ <td align="justify">zero.</td>
+ </tr>
+ <tr>
+ <th align="left">empty products</th>
+ <td align="justify">unity.</td>
+ </tr>
+ </tbody>
+ </table>
+ </div>
+
+ <div class="para" id="p1">
+ <p>The main functions treated in this chapter are the gamma function
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,the psi function
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,the beta function
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>, and the
+ <m:math display="inline">
+ <m:mi>q</m:mi>
+ </m:math>-gamma function
+ <m:math display="inline">
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+ </p>
+ </div>
+
+ <div class="para" id="p2">
+ <p>The notation
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> is due to Legendre. Alternative notations for this function are:
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Π</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> (Gauss) and
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:math>. Alternative notations for the psi function are:
+ </p>
+ </div>
+
+ <div class="table" id="T2">
+ <table align="center">
+ <thead>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ </th>
+ <th align="left">Gauss;
+ <a href="http://dlmf.nist.gov/Contents/bib/J#jahnke:1945:tof";>
+ Jahnke and Emde(1945)
+ </a>
+ </th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>Ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="left">
+ <a href="http://dlmf.nist.gov/Contents/bib/W#whittaker:1927:cma";>
+ Whittaker and Watson(1927)
+ </a>
+ </td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="left">
+ <a href="http://dlmf.nist.gov/Contents/bib/D#davis:1933:thm";>
+ Davis(1933)
+ </a>
+ </td>
+ </tr>
+ <tr>
+ <th align="left">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="sans-serif">F</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ </th>
+ <td align="left">
+ <a href="http://dlmf.nist.gov/Contents/bib/P#pairman:1919:tdt";>
+ Pairman(1919)
+ </a>
+ </td>
+ </tr>
+ </tbody>
+ </table>
+ </div>
+ </div>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfphysicalapplications.xhtml}
+<<dlmfphysicalapplications.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Physical Applications
+ </div>
+ <hr/>
+<h3>Physical Applications</h3>
+
+<p>Suppose the potential energy of a gas of
+ <m:math display="inline">
+ <m:mi>n</m:mi>
+ </m:math> point charges with positions
+ <m:math display="inline">
+ <m:mrow>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:math> and free to move on the infinite line
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo><</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math>, is given by
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>W</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℓ</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:msubsup>
+ <m:mi>x</m:mi>
+ <m:mi mathvariant="normal">ℓ</m:mi>
+ <m:mn>2</m:mn>
+ </m:msubsup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:munder>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi mathvariant="normal">ℓ</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>j</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:munder>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi mathvariant="normal">ℓ</m:mi>
+ </m:msub>
+ <m:mo>-</m:mo>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>j</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>The probability density of the positions when the gas is in thermodynamic
+ equilibrium is:
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>P</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>,</m:mo>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>C</m:mi>
+ <m:mrow>
+ <m:mi>exp</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mi>W</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi>T</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mi>k</m:mi>
+ </m:math> is the Boltzmann constant,
+ <m:math display="inline">
+ <m:mi>T</m:mi>
+ </m:math> the temperature and
+ <m:math display="inline">
+ <m:mi>C</m:mi>
+ </m:math> a constant.
+ Then the partition function (with
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>β</m:mi>
+ <m:mo>=</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mi>T</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:math>) is given by
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>ψ</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>β</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ℝ</m:mi>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:msub>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>β</m:mi>
+ <m:mi>W</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>x</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mi>n</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:msup>
+ <m:mi>β</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mi>n</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:mi>β</m:mi>
+ <m:mi>n</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mn>4</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mo>×</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>β</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>j</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>j</m:mi>
+ <m:mi>β</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>For
+ <m:math display="inline">
+ <m:mi>n</m:mi>
+ </m:math> charges free to move on a circular wire of radius
+ <m:math display="inline">
+ <m:mn>1</m:mn>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mi>W</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:munder>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>≤</m:mo>
+ <m:mi mathvariant="normal">ℓ</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>j</m:mi>
+ <m:mo>≤</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:munder>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mi mathvariant="normal">ℓ</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mi>j</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>and the partition function is given by</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>ψ</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>β</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mo>∫</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>[</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>]</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msup>
+ </m:msub>
+ <m:msup>
+ <m:mi mathvariant="normal">ⅇ</m:mi>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>β</m:mi>
+ <m:mi>W</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:msub>
+ <m:mi>θ</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>n</m:mi>
+ <m:mi>β</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mi>β</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfpolygammafunctions.xhtml}
+<<dlmfpolygammafunctions.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Polygamma Functions
+ </div>
+ <hr/>
+<h3>Polygamma Functions</h3>
+
+<p>The functions
+ <m:math display="inline">
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>n</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>, are called the <em>polygamma functions</em>. In particular,
+ <m:math display="inline">
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> is the <em>trigamma function</em>;
+ <m:math display="inline">
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mi>′</m:mi>
+ <m:mi>′</m:mi>
+ </m:mrow>
+ </m:msup></m:math>,
+ <m:math display="inline">
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msup>
+ </m:math>,
+ <m:math display="inline">
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>4</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msup>
+ </m:math> are the <em>tetra-,</em> <em>penta-,</em> and
+ <em>hexagamma functions</em> respectively. Most properties of these
+ functions follow straightforwardly by differentiation of properties
+ of the psi function. This includes asymptotic expansions.
+</p>
+
+<p>In the second and third equations,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>; for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>n</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>n</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:msup>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>4</m:mn>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>As
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math> in
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mrow>
+ <m:mi>ph</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>≤</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mo>-</m:mo>
+ <m:mi>δ</m:mi>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:none/>
+ <m:mo><</m:mo>
+ <m:mi>π</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∼</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>z</m:mi>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:msub>
+ <m:mi>B</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msub>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfqgammaandbetafunctions.xhtml}
+<<dlmfqgammaandbetafunctions.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- q-Gamma and Beta Functions
+ </div>
+ <hr/>
+<h3>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">q</m:mi>
+ </m:math>-Gamma and Beta Functions
+</h3>
+
+<ul>
+ <li>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">q</m:mi>
+ </m:math>-Factorials</li>
+ <li>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">q</m:mi>
+ </m:math>-Gamma Function</li>
+ <li>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">q</m:mi>
+ </m:math>-Beta Function</li>
+</ul>
+
+<h4>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">q</m:mi>
+ </m:math>-Factorials</h4>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mo>;</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:munderover>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:msup>
+ <m:mi>q</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>,</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">⋯</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>q</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>+</m:mo>
+ <m:msup>
+ <m:mi>q</m:mi>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>q</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mo>;</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>When
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>q</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mo>;</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∏</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:msup>
+ <m:mi>q</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h4>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">q</m:mi></m:math>-Gamma Function</h4>
+
+<p>When
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>q</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac bevelled="true">
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>q</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mo>;</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msub>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mi>q</m:mi>
+ <m:mi>z</m:mi>
+ </m:msup>
+ <m:mspace width="0.2em"/>
+ <m:mo>;</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msub>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:msup>
+ <m:mi>q</m:mi>
+ <m:mi>z</m:mi>
+ </m:msup>
+ </m:mrow>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>Also,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math> is convex for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>, and the analog of the
+ <a href="dlmffunctionrelations.xhtml#bohrmolleruptheorem">
+ Bohr-Mollerup theorem
+ </a> holds.
+</p>
+
+<p>If
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>q</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>r</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>, then
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>r</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>when
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math> or when
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>x</m:mi>
+ <m:mo>></m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:math>, and
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>r</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>when
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>x</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:math>.
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:munder>
+ <m:mo movablelimits="false">lim</m:mo>
+ <m:mrow>
+ <m:mi>q</m:mi>
+ <m:mo>→</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:munder>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>For generalized asymptotic expansions of
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mspace width="0.2em"/>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math> as
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo>→</m:mo>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math> see
+ <a href="http://dlmf.nist.gov/Contents/bib/O#oldedaalhuis:1994:aef";>
+ Olde Daalhuis(1994)
+ </a> and
+ <a href="http://dlmf.nist.gov/Contents/bib/M#moak:1984:tqa";>
+ Moak(1984)
+ </a>.
+</p>
+
+<h4>
+ <m:math display="inline">
+ <m:mi mathvariant="bold-italic">q</m:mi>
+ </m:math>-Beta Function
+</h4>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>a</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>b</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">B</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>,</m:mo>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:msubsup>
+ <m:mo>∫</m:mo>
+ <m:mn>0</m:mn>
+ <m:mn>1</m:mn>
+ </m:msubsup>
+ <m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi>t</m:mi>
+ <m:mrow>
+ <m:mi>a</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>t</m:mi>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mspace width="0.2em"/>
+ <m:mo>;</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:msub>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>t</m:mi>
+ <m:msup>
+ <m:mi>q</m:mi>
+ <m:mi>b</m:mi>
+ </m:msup>
+ </m:mrow>
+ <m:mo>;</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:msub>
+ </m:mfrac>
+ <m:mrow>
+ <m:msub>
+ <m:mi mathvariant="normal">ⅆ</m:mi>
+ <m:mi>q</m:mi>
+ </m:msub>
+ <m:mi>t</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>q</m:mi>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>a</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℜ</m:mi>
+ <m:mi>b</m:mi>
+ </m:mrow>
+ <m:mo>></m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfseriesexpansions.xhtml}
+<<dlmfseriesexpansions.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Series Expansions
+ </div>
+ <hr/>
+<h3>Series Expansions</h3>
+<h6>Contents</h6>
+<ul>
+ <li>Maclaurin Series</li>
+ <li>Other Series</li>
+</ul>
+<h4>Maclaurin Series</h4>
+<p>Throughout this subsection
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math> is
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>where
+ <m:math display="inline">
+ <m:mrow>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+
+ <m:math display="inline">
+ <m:mrow>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mn>2</m:mn>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ </m:math>, and
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>γ</m:mi>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>2</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mn>1</m:mn>
+ </m:msub>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>≥</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<p>For 15D numerical values of
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>c</m:mi>
+ <m:mi>k</m:mi>
+ </m:msub>
+ </m:math> see
+ <a href="http://dlmf.nist.gov/Contents/bib/#abramowitz:1964:hmf";>
+ Abramowitz and Stegun(1964)</a>(p. 256), and
+for 31D values see
+<a href="http://dlmf.nist.gov/Contents/bib/W#wrench:1968:cts";>
+ Wrench(1968)</a>.
+</p>
+
+<a name="equation3"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mi>k</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>,
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>cot</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:msup>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="right">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:mo>|</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>|</m:mo>
+ </m:mrow>
+ <m:mo><</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:math>,
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>±</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>.
+</div>
+
+<p>For 20D numerical values of the coefficients of the Maclaurin series for
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>3</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+</m:math> see
+<a href="http://dlmf.nist.gov/Contents/bib/L#luke:1969:sfa2";>
+ Luke(1969)</a>(p. 299).
+</p>
+
+<p>When
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>≠</m:mo>
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ <m:mo>,</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<a name="equation6"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>z</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>z</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mi>z</m:mi>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>and
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>z</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>z</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mi>z</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>Also,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℑ</m:mi>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mi>y</m:mi>
+ <m:mrow>
+ <m:msup>
+ <m:mi>k</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:mo>+</m:mo>
+ <m:msup>
+ <m:mi>y</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmfsums.xhtml}
+<<dlmfsums.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Sums
+ </div>
+ <hr/>
+<h3>Sums</h3>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mi>k</m:mi>
+ </m:msup>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mi>k</m:mi>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:msup>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:mn>8</m:mn>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>k</m:mi>
+ </m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>ζ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>3</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:msup>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mi>′</m:mi>
+ <m:mi>′</m:mi>
+ </m:mrow>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>For further sums involving the psi function see
+<a href="http://dlmf.nist.gov/Contents/bib/H#hansen:1975:tsp";>
+ Hansen(1975)
+</a>(pp. 360–367). For sums of gamma functions see
+<a href="http://dlmf.nist.gov/Contents/bib/#andrews:1999:sfu";>
+ Andrews <em>et.al.</em>(1999)
+</a>(Chapters 2 and 3).
+</p>
+
+<p>For related sums involving finite field analogs of the gamma and
+beta functions (Gauss and Jacobi sums) see
+<a href="http://dlmf.nist.gov/Contents/bib/#andrews:1999:sfu";>
+ Andrews <em>et.al.</em>(1999)
+</a>(Chapter 1) and
+<a href="http://dlmf.nist.gov/Contents/bib/T#terras:1999:fao";>
+ Terras(1999)
+</a>.
+</p>
+<<page foot>>
+@
+
+\subsection{dlmfsoftware.xhtml}
+<<dlmfsoftware.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Software
+ </div>
+ <hr/>
+<<page foot>>
+@
+
+\subsection{dlmfspecialvaluesandextrema.xhtml}
+<<dlmfspecialvaluesandextrema.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Special Values and Extrema
+ </div>
+ <hr/>
+<h3>Special Values and Extrema</h3>
+<h6>Contents</h6>
+ <ul>
+ <li>Gamma Function</li>
+ <li>Psi Function</li>
+ <li>Extrema</li>
+ </ul>
+
+<h4>Gamma Function</h4>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">!</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>∣</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∣</m:mo>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mi>y</m:mi>
+ <m:mrow>
+ <m:mi>sinh</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:msup>
+ <m:mrow>
+ <m:mo>∣</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>∣</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mi>cosh</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>4</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>3</m:mn>
+ <m:mn>4</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:msqrt>
+ <m:mn>2</m:mn>
+ </m:msqrt>
+ </m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>cosh</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mrow>
+ <m:mi>sinh</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:msup>
+ <m:mi>π</m:mi>
+ <m:mfrac bevelled="true">
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:msup>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1.77245 38509 05516 02729</m:mn>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>2.67893 85347 07747 63365</m:mn>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>2</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1.35411 79394 26400 41694</m:mn>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>4</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>3.62560 99082 21908 31193</m:mn>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>3</m:mn>
+ <m:mn>4</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mn>1.22541 67024 65177 64512</m:mn>
+ <m:mi mathvariant="normal">…</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h4>Psi Function</h4>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mn>1</m:mn>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<a name="equation13"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mi>n</m:mi>
+ </m:munderover>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>k</m:mi>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mn>2</m:mn>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>3</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>+</m:mo>
+ <m:mi mathvariant="normal">…</m:mi>
+ <m:mo>+</m:mo>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:mfrac>
+ </m:mstyle>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mo>≥</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℑ</m:mi>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>coth</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℑ</m:mi>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mstyle displaystyle="false">
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ </m:mstyle>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>tanh</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi mathvariant="normal">ℑ</m:mi>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>1</m:mn>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi mathvariant="normal">ⅈ</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>coth</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>y</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<p>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mn>0</m:mn>
+ <m:mo><</m:mo>
+ <m:mi>p</m:mi>
+ <m:mo><</m:mo>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ </m:math> are integers, then
+</p>
+
+<a name="equation19"/>
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>p</m:mi>
+ <m:mi>q</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>γ</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mi>q</m:mi>
+ </m:mrow>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>cot</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mi>π</m:mi>
+ <m:mi>p</m:mi>
+ </m:mrow>
+ <m:mi>q</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mn>2</m:mn>
+ </m:mfrac>
+ <m:mrow>
+ <m:munderover>
+ <m:mo movablelimits="false">∑</m:mo>
+ <m:mrow>
+ <m:mi>k</m:mi>
+ <m:mo>=</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>q</m:mi>
+ <m:mo>-</m:mo>
+ <m:mn>1</m:mn>
+ </m:mrow>
+ </m:munderover>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi>k</m:mi>
+ <m:mi>p</m:mi>
+ </m:mrow>
+ <m:mi>q</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mo>-</m:mo>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mrow>
+ <m:mi>cos</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mrow>
+ <m:mn>2</m:mn>
+ <m:mi>π</m:mi>
+ <m:mi>k</m:mi>
+ </m:mrow>
+ <m:mi>q</m:mi>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+
+<h4>Extrema</h4>
+<div>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mrow>
+ <m:msup>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mo>′</m:mo>
+ </m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mi>ψ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>=</m:mo>
+ <m:mn>0</m:mn>
+ </m:mrow>
+ </m:math>.
+</div>
+<br/>
+<div class="center">
+ <table align="center">
+ <thead>
+ <tr>
+ <th align="center" class="b l r t">
+ <m:math display="inline">
+ <m:mi>n</m:mi>
+ </m:math>
+ </th>
+ <th align="center" class="b r t">
+ <m:math display="inline">
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ </m:math>
+ </th>
+ <th align="center" class="b r t">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi mathvariant="normal">Γ</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+ </th>
+ </tr>
+ </thead>
+ <tbody>
+ <tr>
+ <th align="right" class="l r">0
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mn>1.46163 21449</m:mn>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mn>0.88560 31944</m:mn>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="l r">1
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>0.50408 30083</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>3.54464 36112</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="l r">2
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>1.57349 84732</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mn>2.30240 72583</m:mn>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="B l r">3
+ </th>
+ <td align="right" class="B r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>2.61072 08875</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="B r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>0.88813 63584</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="l r">4
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>3.63529 33665</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mn>0.24512 75398</m:mn>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="l r">5
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>4.65323 77626</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>0.05277 96396</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="B l r">6
+ </th>
+ <td align="right" class="B r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>5.66716 24513</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="B r">
+ <m:math display="inline">
+ <m:mn>0.00932 45945</m:mn>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="l r">7
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>6.67841 82649</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>0.00139 73966</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="l r">8
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>7.68778 83250</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mn>0.00018 18784</m:mn>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="l r">9
+ </th>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>8.69576 41633</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>0.00002 09253</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ </tr>
+ <tr>
+ <th align="right" class="b l r">10
+ </th>
+ <td align="right" class="b r">
+ <m:math display="inline">
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mn>9.70267 25406</m:mn>
+ </m:mrow>
+ </m:math>
+ </td>
+ <td align="right" class="b r">
+ <m:math display="inline">
+ <m:mn>0.00000 21574</m:mn>
+ </m:math>
+ </td>
+ </tr>
+ </tbody>
+ </table>
+</div>
+
+<p>As
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:mi mathvariant="normal">∞</m:mi>
+ </m:mrow>
+ </m:math>,
+</p>
+
+<div align="center">
+ <m:math display="block">
+ <m:mrow>
+ <m:mrow>
+ <m:msub>
+ <m:mi>x</m:mi>
+ <m:mi>n</m:mi>
+ </m:msub>
+ <m:mo>=</m:mo>
+ <m:mrow>
+ <m:mrow>
+ <m:mo>-</m:mo>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mi>π</m:mi>
+ </m:mfrac>
+ <m:mrow>
+ <m:mi>arctan</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mi>π</m:mi>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ <m:mo>+</m:mo>
+ <m:mrow>
+ <m:mi>O</m:mi>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mfrac>
+ <m:mn>1</m:mn>
+ <m:mrow>
+ <m:mi>n</m:mi>
+ <m:msup>
+ <m:mrow>
+ <m:mo>(</m:mo>
+ <m:mrow>
+ <m:mi>ln</m:mi>
+ <m:mi>n</m:mi>
+ </m:mrow>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ <m:mn>2</m:mn>
+ </m:msup>
+ </m:mrow>
+ </m:mfrac>
+ <m:mo>)</m:mo>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:mrow>
+ </m:math>
+</div>
+<<page foot>>
+@
+
+\subsection{dlmftables.xhtml}
+<<dlmftables.xhtml>>=
+<<standard head>>
+ </head>
+ <body>
+<<page head>>
+ <div align="center">
+ <a href="http://dlmf.nist.gov";>
+ Digital Library of Mathematical Functions
+ </a><br/>
+ The Gamma Function -- Tables
+ </div>
+ <hr/>
+<h3>Tables</h3>
+
+These tables show Axiom's compliance with published standard values.
+In all cases shown here Axiom conforms to the accuracy of the published
+tables.
+
+<ul>
+ <li>The Gamma Function</li>
+ <li>The Psi Function</li>
+</ul>
+
+<h4>The Gamma Function</h4>
+
+This table was constructed from the published values in the
+Handbook of Mathematical Functions, by Milton Abramowitz
+and Irene A. Stegun, by Dover (1965), pp 267-270.
+
+The first column is the point where the Gamma function is evaluated.
+The second column is the value reported in the Handbook.
+The third column is the actual value computed by Axiom at the given point.
+The fourth column is the difference of Axiom's value and the Handbook value.
+
+<table border="1">
+ <tr>
+ <th>point</th>
+ <th>Handbook Value</th>
+ <th>Axiom Computed Value</th>
+ <th>Difference</th>
+ </tr>
+ <tr>
+ <td>1.000</td>
+ <td>1.0000000000</td>
+ <td>1.</td>
+ <td align="right">0.</td>
+ </tr>
+ <tr>
+ <td>1.005</td>
+ <td>0.9971385354</td>
+ <td>0.9971385352483757</td>
+ <td align="right">-1.51E-10</td>
+ </tr>
+ <tr>
+ <td>1.010</td>
+ <td>0.9943258512</td>
+ <td>0.99432585118631189</td>
+ <td align="right">-2.03E-11</td>
+ </tr>
+ <tr>
+ <td>1.015</td>
+ <td>0.9915612888</td>
+ <td>0.99156128884131323</td>
+ <td align="right">4.14E-11</td>
+ </tr>
+ <tr>
+ <td>1.020</td>
+ <td>0.9888442033</td>
+ <td>0.9888442032538789</td>
+ <td align="right">-4.31E-11</td>
+ </tr>
+ <tr>
+ <td>1.025</td>
+ <td>0.9861739633</td>
+ <td>0.98617396313592742</td>
+ <td align="right">-1.54E-10</td>
+ </tr>
+ <tr>
+ <td>1.030</td>
+ <td>0.9835499506</td>
+ <td>0.98354995053928918</td>
+ <td align="right">-7.59E-11</td>
+ </tr>
+ <tr>
+ <td>1.035</td>
+ <td>0.9809715606</td>
+ <td>0.98097156056367696</td>
+ <td align="right">-4.60E-11</td>
+ </tr>
+ <tr>
+ <td>1.040</td>
+ <td>0.9784382009</td>
+ <td>0.9784382009247683</td>
+ <td align="right"> 3.00E-11</td>
+ </tr>
+ <tr>
+ <td>1.045</td>
+ <td>0.9759492919</td>
+ <td>0.97594929183099266</td>
+ <td align="right">-6.55E-11</td>
+ </tr>
+ <tr>
+ <td>1.050</td>
+ <td>0.9735042656</td>
+ <td>0.97350426556841785</td>
+ <td align="right">-2.72E-11</td>
+ </tr>
+ <tr>
+ <td>1.055</td>
+ <td>0.9711025663</td>
+ <td>0.97110256624499502</td>
+ <td align="right">-6.77E-11</td>
+ </tr>
+ <tr>
+ <td>1.060</td>
+ <td>0.9687436495</td>
+ <td>0.96874364951272707</td>
+ <td align="right">-2.36E-12</td>
+ </tr>
+ <tr>
+ <td>1.065</td>
+ <td>0.9664269823</td>
+ <td>0.96642698229777113</td>
+ <td align="right">-1.37E-11</td>
+ </tr>
+ <tr>
+ <td>1.070</td>
+ <td>0.9641520425</td>
+ <td>0.96415204253821729</td>
+ <td align="right"> 4.61E-11</td>
+ </tr>
+ <tr>
+ <td>1.075</td>
+ <td>0.9619183189</td>
+ <td>0.96191831892929192</td>
+ <td align="right"> 2.31E-11</td>
+ </tr>
+ <tr>
+ <td>1.080</td>
+ <td>0.9597253107</td>
+ <td>0.95972531067573963</td>
+ <td align="right">-3.00E-11</td>
+ </tr>
+ <tr>
+ <td>1.085</td>
+ <td>0.9575725273</td>
+ <td>0.95757252725116249</td>
+ <td align="right">-3.68E-11</td>
+ </tr>
+ <tr>
+ <td>1.090</td>
+ <td>0.9554594882</td>
+ <td>0.95545948816407866</td>
+ <td align="right">-4.24E-11</td>
+ </tr>
+ <tr>
+ <td>1.095</td>
+ <td>0.9533857227</td>
+ <td>0.95338572273049704</td>
+ <td align="right"> 2.34E-11</td>
+ </tr>
+ <tr>
+ <td>1.100</td>
+ <td>0.9513507699</td>
+ <td>0.95135076987625944</td>
+ <td align="right">-2.49E-11</td>
+ </tr>
+ <tr>
+ <td>1.105</td>
+ <td>0.9493541778</td>
+ <td>0.94935417782771081</td>
+ <td align="right"> 2.11E-11</td>
+ </tr>
+ <tr>
+ <td>1.110</td>
+ <td>0.9473955040</td>
+ <td>0.94739550404472173</td>
+ <td align="right"> 5.80E-11</td>
+ </tr>
+ <tr>
+ <td>1.115</td>
+ <td>0.9454743149</td>
+ <td>0.94547431492209555</td>
+ <td align="right"> 1.12E-11</td>
+ </tr>
+ <tr>
+ <td>1.120</td>
+ <td>0.9435901856</td>
+ <td>0.94359018561564112</td>
+ <td align="right"> 1.06E-11</td>
+ </tr>
+ <tr>
+ <td>1.125</td>
+ <td>0.9417426997</td>
+ <td>0.94174269984970138</td>
+ <td align="right"> 1.39E-10</td>
+ </tr>
+ <tr>
+ <td>1.130</td>
+ <td>0.9399314497</td>
+ <td>0.93993144972988807</td>
+ <td align="right"> 1.67E-11</td>
+ </tr>
+ <tr>
+ <td>1.135</td>
+ <td>0.9381560356</td>
+ <td>0.93815603556085947</td>
+ <td align="right">-5.14E-11</td>
+ </tr>
+ <tr>
+ <td>1.140</td>
+ <td>0.9364160657</td>
+ <td>0.93641606566898694</td>
+ <td align="right">-2.97E-11</td>
+ </tr>
+ <tr>
+ <td>1.145</td>
+ <td>0.9347111562</td>
+ <td>0.93471115622975964</td>
+ <td align="right"> 2.05E-11</td>
+ </tr>
+ <tr>
+ <td>1.150</td>
+ <td>0.9330409311</td>
+ <td>0.93304093109978414</td>
+ <td align="right"> 6.51E-12</td>
+ </tr>
+ <tr>
+ <td>1.155</td>
+ <td>0.9314050217</td>
+ <td>0.93140502165323868</td>
+ <td align="right">-3.93E-11</td>
+ </tr>
+ <tr>
+ <td>1.160</td>
+ <td>0.9298030666</td>
+ <td>0.92980306664109957</td>
+ <td align="right"> 4.51E-11</td>
+ </tr>
+ <tr>
+ <td>1.165</td>
+ <td>0.9282347120</td>
+ <td>0.92823471196190366</td>
+ <td align="right">-2.59E-11</td>
+ </tr>
+ <tr>
+ <td>1.170</td>
+ <td>0.9266996106</td>
+ <td>0.92669961062266581</td>
+ <td align="right"> 2.10E-11</td>
+ </tr>
+ <tr>
+ <td>1.175</td>
+ <td>0.9251974225</td>
+ <td>0.92519742251686099</td>
+ <td align="right"> 1.24E-11</td>
+ </tr>
+ <tr>
+ <td>1.180</td>
+ <td>0.9237278143</td>
+ <td>0.92372781430006712</td>
+ <td align="right">-1.17E-11</td>
+ </tr>
+ <tr>
+ <td>1.185</td>
+ <td>0.9222904591</td>
+ <td>0.92229045925047382</td>
+ <td align="right"> 1.49E-10</td>
+ </tr>
+ <tr>
+ <td>1.190</td>
+ <td>0.9208850371</td>
+ <td>0.92088503713299241</td>
+ <td align="right"> 2.60E-11</td>
+ </tr>
+ <tr>
+ <td>1.195</td>
+ <td>0.9195112341</td>
+ <td>0.91951123406686597</td>
+ <td align="right">-2.98E-11</td>
+ </tr>
+ <tr>
+ <td>1.200</td>
+ <td>0.9181687424</td>
+ <td>0.91816874239667101</td>
+ <td align="right">-1.67E-11</td>
+ </tr>
+ <tr>
+ <td>1.205</td>
+ <td>0.9168572606</td>
+ <td>0.91685726056661909</td>
+ <td align="right">-3.28E-11</td>
+ </tr>
+ <tr>
+ <td>1.210</td>
+ <td>0.9155764930</td>
+ <td>0.91557649299805532</td>
+ <td align="right"> 8.85E-12</td>
+ </tr>
+ <tr>
+ <td>1.215</td>
+ <td>0.9143261400</td>
+ <td>0.91432614997006778</td>
+ <td align="right"> 9.98E-9</td>
+ </tr>
+ <tr>
+ <td>1.220</td>
+ <td>0.9131059475</td>
+ <td>0.91310594750311536</td>
+ <td align="right"> 1.37E-11</td>
+ </tr>
+ <tr>
+ <td>1.225</td>
+ <td>0.9119156071</td>
+ <td>0.91191560725927312</td>
+ <td align="right"> 1.49E-10</td>
+ </tr>
+ <tr>
+ <td>1.230</td>
+ <td>0.9107548564</td>
+ <td>0.91075485637655895</td>
+ <td align="right">-1.50E-11</td>
+ </tr>
+ <tr>
+ <td>1.235</td>
+ <td>0.9096234274</td>
+ <td>0.90962342744425173</td>
+ <td align="right"> 4.03E-11</td>
+ </tr>
+ <tr>
+ <td>1.240</td>
+ <td>0.9085210583</td>
+ <td>0.90852105834198582</td>
+ <td align="right"> 4.21E-11</td>
+ </tr>
+ <tr>
+ <td>1.245</td>
+ <td>0.9074474922</td>
+ <td>0.90744749215126341</td>
+ <td align="right">-5.77E-11</td>
+ </tr>
+ <tr>
+ <td>1.250</td>
+ <td>0.9064024771</td>
+ <td>0.90640247705547716</td>
+ <td align="right">-3.68E-11</td>
+ </tr>
+ <tr>
+ <td>1.255</td>
+ <td>0.9053857663</td>
+ <td>0.90538576624240463</td>
+ <td align="right">-5.23E-11</td>
+ </tr>
+ <tr>
+ <td>1.260</td>
+ <td>0.9043971178</td>
+ <td>0.90439711780910215</td>
+ <td align="right"> 2.01E-11</td>
+ </tr>
+ <tr>
+ <td>1.265</td>
+ <td>0.9034362946</td>
+ <td>0.90343629466913566</td>
+ <td align="right"> 5.78E-11</td>
+ </tr>
+ <tr>
+ <td>1.270</td>
+ <td>0.9025030645</td>
+ <td>0.90250306446208062</td>
+ <td align="right">-5.13E-11</td>
+ </tr>
+ <tr>
+ <td>1.275</td>
+ <td>0.9015971994</td>
+ <td>0.90159719946523187</td>
+ <td align="right"> 5.66E-11</td>
+ </tr>
+ <tr>
+ <td>1.280</td>
+ <td>0.9007184765</td>
+ <td>0.90071847650745973</td>
+ <td align="right"> 5.78E-13</td>
+ </tr>
+ <tr>
+ <td>1.285</td>
+ <td>0.8998666769</td>
+ <td>0.89986667689491762</td>
+ <td align="right"> 5.55E-12</td>
+ </tr>
+ <tr>
+ <td>1.290</td>
+ <td>0.8990415863</td>
+ <td>0.89904158628967101</td>
+ <td align="right">-3.93E-12</td>
+ </tr>
+ <tr>
+ <td>1.295</td>
+ <td>0.8982429947</td>
+ <td>0.89824299468914737</td>
+ <td align="right">-1.72E-11</td>
+ </tr>
+ <tr>
+ <td>1.300</td>
+ <td>0.8974706963</td>
+ <td>0.89747069630804477</td>
+ <td align="right"> 2.65E-12</td>
+ </tr>
+ <tr>
+ <td>1.305</td>
+ <td>0.8967244895</td>
+ <td>0.89672448951215833</td>
+ <td align="right"> 2.37E-11</td>
+ </tr>
+ <tr>
+ <td>1.310</td>
+ <td>0.8960041767</td>
+ <td>0.89600417674396082</td>
+ <td align="right"> 4.53E-11</td>
+ </tr>
+ <tr>
+ <td>1.315</td>
+ <td>0.8953095644</td>
+ <td>0.89530956444995535</td>
+ <td align="right"> 5.43E-11</td>
+ </tr>
+ <tr>
+ <td>1.320</td>
+ <td>0.8946404630</td>
+ <td>0.89464046300975775</td>
+ <td align="right"> 1.28E-11</td>
+ </tr>
+ <tr>
+ <td>1.325</td>
+ <td>0.8939966866</td>
+ <td>0.89399668666686083</td>
+ <td align="right"> 7.95E-11</td>
+ </tr>
+ <tr>
+ <td>1.330</td>
+ <td>0.8933780535</td>
+ <td>0.89337805346103716</td>
+ <td align="right">-3.97E-11</td>
+ </tr>
+ <tr>
+ <td>1.335</td>
+ <td>0.8927843850</td>
+ <td>0.89278438516233538</td>
+ <td align="right"> 1.51E-10</td>
+ </tr>
+ <tr>
+ <td>1.340</td>
+ <td>0.8922155072</td>
+ <td>0.89221550720663356</td>
+ <td align="right"> 1.43E-11</td>
+ </tr>
+ <tr>
+ <td>1.345</td>
+ <td>0.8916712485</td>
+ <td>0.89167124863270442</td>
+ <td align="right"> 1.24E-10</td>
+ </tr>
+ <tr>
+ <td>1.350</td>
+ <td>0.8911514420</td>
+ <td>0.89115144202666452</td>
+ <td align="right"> 3.78E-11</td>
+ </tr>
+ <tr>
+ <td>1.355</td>
+ <td>0.8906559235</td>
+ <td>0.89065592343803057</td>
+ <td align="right">-5.12E-11</td>
+ </tr>
+ <tr>
+ <td>1.360</td>
+ <td>0.8901845324</td>
+ <td>0.8901845323574008</td>
+ <td align="right">-5.70E-11</td>
+ </tr>
+ <tr>
+ <td>1.365</td>
+ <td>0.8897371116</td>
+ <td>0.88973711163470881</td>
+ <td align="right"> 3.11E-11</td>
+ </tr>
+ <tr>
+ <td>1.370</td>
+ <td>0.8893135074</td>
+ <td>0.88931350742948501</td>
+ <td align="right"> 4.09E-11</td>
+ </tr>
+ <tr>
+ <td>1.375</td>
+ <td>0.8889135692</td>
+ <td>0.88891356915622532</td>
+ <td align="right">-5.89E-11</td>
+ </tr>
+ <tr>
+ <td>1.380</td>
+ <td>0.8885371494</td>
+ <td>0.88853714943101736</td>
+ <td align="right"> 2.03E-11</td>
+ </tr>
+ <tr>
+ <td>1.385</td>
+ <td>0.8881841041</td>
+ <td>0.88818410401940351</td>
+ <td align="right">-9.53E-11</td>
+ </tr>
+ <tr>
+ <td>1.390</td>
+ <td>0.8878542918</td>
+ <td>0.88785429178544073</td>
+ <td align="right">-1.00E-11</td>
+ </tr>
+ <tr>
+ <td>1.395</td>
+ <td>0.8875475748</td>
+ <td>0.88754757464193323</td>
+ <td align="right">-1.49E-10</td>
+ </tr>
+ <tr>
+ <td>1.400</td>
+ <td>0.8872638175</td>
+ <td>0.88726381750180738</td>
+ <td align="right">-7.13E-12</td>
+ </tr>
+ <tr>
+ <td>1.405</td>
+ <td>0.8870028884</td>
+ <td>0.88700288823059736</td>
+ <td align="right">-1.66E-10</td>
+ </tr>
+ <tr>
+ <td>1.410</td>
+ <td>0.8867646576</td>
+ <td>0.88676465760002188</td>
+ <td align="right"> 3.66E-12</td>
+ </tr>
+ <tr>
+ <td>1.415</td>
+ <td>0.8865489993</td>
+ <td>0.88654899924499497</td>
+ <td align="right">-4.45E-11</td>
+ </tr>
+ <tr>
+ <td>1.420</td>
+ <td>0.8863557896</td>
+ <td>0.88635578960951567</td>
+ <td align="right">-1.60E-12</td>
+ </tr>
+ <tr>
+ <td>1.425</td>
+ <td>0.8861849081</td>
+ <td>0.88618490791840432</td>
+ <td align="right">-1.81E-10</td>
+ </tr>
+ <tr>
+ <td>1.430</td>
+ <td>0.8860362361</td>
+ <td>0.88603623612466142</td>
+ <td align="right"> 2.35E-11</td>
+ </tr>
+ <tr>
+ <td>1.435</td>
+ <td>0.8859096587</td>
+ <td>0.88590965887072826</td>
+ <td align="right"> 1.59E-10</td>
+ </tr>
+ <tr>
+ <td>1.440</td>
+ <td>0.8858050635</td>
+ <td>0.88580506344804788</td>
+ <td align="right">-5.45E-11</td>
+ </tr>
+ <tr>
+ <td>1.445</td>
+ <td>0.8857223397</td>
+ <td>0.88572233975753722</td>
+ <td align="right"> 5.12E-11</td>
+ </tr>
+ <tr>
+ <td>1.450</td>
+ <td>0.8856613803</td>
+ <td>0.88566138027095553</td>
+ <td align="right">-3.63E-11</td>
+ </tr>
+ <tr>
+ <td>1.455</td>
+ <td>0.8856220700</td>
+ <td>0.88562207999314335</td>
+ <td align="right"> 9.99E-9</td>
+ </tr>
+ <tr>
+ <td>1.460</td>
+ <td>0.8856043364</td>
+ <td>0.88560433642511449</td>
+ <td align="right"> 3.29E-11</td>
+ </tr>
+ <tr>
+ <td>1.465</td>
+ <td>0.8856080495</td>
+ <td>0.88560804952797856</td>
+ <td align="right"> 4.00E-11</td>
+ </tr>
+ <tr>
+ <td>1.470</td>
+ <td>0.8856331217</td>
+ <td>0.88563312168767672</td>
+ <td align="right">-2.25E-11</td>
+ </tr>
+ <tr>
+ <td>1.475</td>
+ <td>0.8856794575</td>
+ <td>0.88567945767984679</td>
+ <td align="right"> 1.68E-10</td>
+ </tr>
+ <tr>
+ <td>1.480</td>
+ <td>0.8857469646</td>
+ <td>0.88574696463853297</td>
+ <td align="right"> 3.58E-11</td>
+ </tr>
+ <tr>
+ <td>1.485</td>
+ <td>0.8858355520</td>
+ <td>0.88583555202000774</td>
+ <td align="right"> 1.39E-11</td>
+ </tr>
+ <tr>
+ <td>1.490</td>
+ <td>0.8859451316</td>
+ <td>0.885945131572484</td>
+ <td align="right">-2.22E-11</td>
+ </tr>
+ <tr>
+ <td>1.495</td>
+ <td>0.8860756174</td>
+ <td>0.88607561730422169</td>
+ <td align="right">-9.20E-11</td>
+ </tr>
+ <tr>
+ <td>1.500</td>
+ <td>0.8862269255</td>
+ <td>0.88622692545275816</td>
+ <td align="right">-5.14E-11</td>
+ </tr>
+ <tr>
+ <td>1.505</td>
+ <td>0.8863989744</td>
+ <td>0.88639897445482596</td>
+ <td align="right"> 5.62E-11</td>
+ </tr>
+ <tr>
+ <td>1.510</td>
+ <td>0.8865916850</td>
+ <td>0.88659168491694862</td>
+ <td align="right">-8.75E-11</td>
+ </tr>
+ <tr>
+ <td>1.515</td>
+ <td>0.8868049797</td>
+ <td>0.88680497958669369</td>
+ <td align="right">-1.15E-10</td>
+ </tr>
+ <tr>
+ <td>1.520</td>
+ <td>0.8870387833</td>
+ <td>0.88703878332457031</td>
+ <td align="right"> 3.78E-11</td>
+ </tr>
+ <tr>
+ <td>1.525</td>
+ <td>0.8872930231</td>
+ <td>0.88729302307655866</td>
+ <td align="right">-3.89E-11</td>
+ </tr>
+ <tr>
+ <td>1.530</td>
+ <td>0.8875676278</td>
+ <td>0.88756762784725507</td>
+ <td align="right"> 5.05E-11</td>
+ </tr>
+ <tr>
+ <td>1.535</td>
+ <td>0.8878625287</td>
+ <td>0.88786252867361892</td>
+ <td align="right">-2.97E-11</td>
+ </tr>
+ <tr>
+ <td>1.540</td>
+ <td>0.8881776586</td>
+ <td>0.88817765859552456</td>
+ <td align="right">-1.03E-11</td>
+ </tr>
+ <tr>
+ <td>1.545</td>
+ <td>0.8885129527</td>
+ <td>0.88851295264558472</td>
+ <td align="right">-4.41E-11</td>
+ </tr>
+ <tr>
+ <td>1.550</td>
+ <td>0.8888683478</td>
+ <td>0.88886834780261559</td>
+ <td align="right"> 2.74E-12</td>
+ </tr>
+ <tr>
+ <td>1.555</td>
+ <td>0.8892437830</td>
+ <td>0.88924378298210571</td>
+ <td align="right">-1.06E-11</td>
+ </tr>
+ <tr>
+ <td>1.560</td>
+ <td>0.8896391990</td>
+ <td>0.88963919900923583</td>
+ <td align="right">-3.65E-12</td>
+ </tr>
+ <tr>
+ <td>1.565</td>
+ <td>0.8900545387</td>
+ <td>0.89005453859597561</td>
+ <td align="right">-1.04E-10</td>
+ </tr>
+ <tr>
+ <td>1.570</td>
+ <td>0.8904897463</td>
+ <td>0.89048974631869759</td>
+ <td align="right"> 2.61E-11</td>
+ </tr>
+ <tr>
+ <td>1.575</td>
+ <td>0.8909447686</td>
+ <td>0.89094476859629979</td>
+ <td align="right"> 8.93E-12</td>
+ </tr>
+ <tr>
+ <td>1.580</td>
+ <td>0.8914195537</td>
+ <td>0.89141955366882042</td>
+ <td align="right">-2.38E-11</td>
+ </tr>
+ <tr>
+ <td>1.585</td>
+ <td>0.8919140515</td>
+ <td>0.8919140515765388</td>
+ <td align="right"> 8.47E-11</td>
+ </tr>
+ <tr>
+ <td>1.590</td>
+ <td>0.8924282141</td>
+ <td>0.8924282141395512</td>
+ <td align="right"> 3.07E-11</td>
+ </tr>
+ <tr>
+ <td>1.595</td>
+ <td>0.8929619949</td>
+ <td>0.89296199493781103</td>
+ <td align="right"> 4.74E-11</td>
+ </tr>
+ <tr>
+ <td>1.600</td>
+ <td>0.8935153493</td>
+ <td>0.89351534928506793</td>
+ <td align="right">-2.24E-11</td>
+ </tr>
+ <tr>
+ <td>1.605</td>
+ <td>0.8940882342</td>
+ <td>0.89408823423580575</td>
+ <td align="right"> 3.63E-11</td>
+ </tr>
+ <tr>
+ <td>1.610</td>
+ <td>0.8946806085</td>
+ <td>0.89468060852796683</td>
+ <td align="right"> 2.74E-11</td>
+ </tr>
+ <tr>
+ <td>1.615</td>
+ <td>0.8952924327</td>
+ <td>0.89529243259029823</td>
+ <td align="right">-9.74E-11</td>
+ </tr>
+ <tr>
+ <td>1.620</td>
+ <td>0.8959236685</td>
+ <td>0.89592366851824745</td>
+ <td align="right"> 2.86E-11</td>
+ </tr>
+ <tr>
+ <td>1.625</td>
+ <td>0.8965742800</td>
+ <td>0.89657428005659789</td>
+ <td align="right"> 6.46E-11</td>
+ </tr>
+ <tr>
+ <td>1.630</td>
+ <td>0.8972442326</td>
+ <td>0.89724423258250552</td>
+ <td align="right">-7.80E-12</td>
+ </tr>
+ <tr>
+ <td>1.635</td>
+ <td>0.8979334930</td>
+ <td>0.89793349308892934</td>
+ <td align="right"> 9.89E-11</td>
+ </tr>
+ <tr>
+ <td>1.640</td>
+ <td>0.8986420302</td>
+ <td>0.89864203016845012</td>
+ <td align="right">-2.68E-11</td>
+ </tr>
+ <tr>
+ <td>1.645</td>
+ <td>0.8993698138</td>
+ <td>0.89936981399746452</td>
+ <td align="right"> 2.04E-10</td>
+ </tr>
+ <tr>
+ <td>1.650</td>
+ <td>0.9001168163</td>
+ <td>0.9001168163207548</td>
+ <td align="right"> 1.21E-11</td>
+ </tr>
+ <tr>
+ <td>1.655</td>
+ <td>0.9008830104</td>
+ <td>0.90088301043641827</td>
+ <td align="right"> 2.24E-11</td>
+ </tr>
+ <tr>
+ <td>1.660</td>
+ <td>0.9016683712</td>
+ <td>0.90166837118115595</td>
+ <td align="right">-1.49E-11</td>
+ </tr>
+ <tr>
+ <td>1.665</td>
+ <td>0.9024728748</td>
+ <td>0.90247287490643413</td>
+ <td align="right"> 1.16E-10</td>
+ </tr>
+ <tr>
+ <td>1.670</td>
+ <td>0.9032964995</td>
+ <td>0.9032964995021503</td>
+ <td align="right">-1.09E-11</td>
+ </tr>
+ <tr>
+ <td>1.675</td>
+ <td>0.9041392243</td>
+ <td>0.90413922432675797</td>
+ <td align="right"> 3.24E-11</td>
+ </tr>
+ <tr>
+ <td>1.680</td>
+ <td>0.9050010302</td>
+ <td>0.90500103023115419</td>
+ <td align="right"> 4.40E-11</td>
+ </tr>
+ <tr>
+ <td>1.685</td>
+ <td>0.9058818996</td>
+ <td>0.90588189953639731</td>
+ <td align="right">-7.63E-11</td>
+ </tr>
+ <tr>
+ <td>1.690</td>
+ <td>0.9067818160</td>
+ <td>0.90678181602099839</td>
+ <td align="right"> 9.93E-12</td>
+ </tr>
+ <tr>
+ <td>1.695</td>
+ <td>0.9077007650</td>
+ <td>0.90770076490852225</td>
+ <td align="right">-9.63E-11</td>
+ </tr>
+ <tr>
+ <td>1.700</td>
+ <td>0.9086387329</td>
+ <td>0.90863873285549646</td>
+ <td align="right">-5.97E-11</td>
+ </tr>
+ <tr>
+ <td>1.705</td>
+ <td>0.9095957079</td>
+ <td>0.90959570793962097</td>
+ <td align="right"> 4.25E-11</td>
+ </tr>
+ <tr>
+ <td>1.710</td>
+ <td>0.9105716796</td>
+ <td>0.9105716796482709</td>
+ <td align="right"> 5.89E-11</td>
+ </tr>
+ <tr>
+ <td>1.715</td>
+ <td>0.9115666390</td>
+ <td>0.91156663886729161</td>
+ <td align="right">-1.31E-10</td>
+ </tr>
+ <tr>
+ <td>1.720</td>
+ <td>0.9125805779</td>
+ <td>0.91258057787007674</td>
+ <td align="right">-1.93E-11</td>
+ </tr>
+ <tr>
+ <td>1.725</td>
+ <td>0.9136134904</td>
+ <td>0.91361349029479011</td>
+ <td align="right">-1.16E-10</td>
+ </tr>
+ <tr>
+ <td>1.730</td>
+ <td>0.9146653712</td>
+ <td>0.91466537118231861</td>
+ <td align="right">-2.63E-11</td>
+ </tr>
+ <tr>
+ <td>1.735</td>
+ <td>0.9157362171</td>
+ <td>0.9157362168940244</td>
+ <td align="right">-2.15E-10</td>
+ </tr>
+ <tr>
+ <td>1.740</td>
+ <td>0.9168260252</td>
+ <td>0.91682602514979106</td>
+ <td align="right">-5.47E-11</td>
+ </tr>
+ <tr>
+ <td>1.745</td>
+ <td>0.9179347950</td>
+ <td>0.91793479500653363</td>
+ <td align="right"> 8.97E-12</td>
+ </tr>
+ <tr>
+ <td>1.750</td>
+ <td>0.9190625268</td>
+ <td>0.91906252684888312</td>
+ <td align="right"> 3.95E-11</td>
+ </tr>
+ <tr>
+ <td>1.755</td>
+ <td>0.9202092224</td>
+ <td>0.92020922238011904</td>
+ <td align="right">-3.48E-11</td>
+ </tr>
+ <tr>
+ <td>1.760</td>
+ <td>0.9213748846</td>
+ <td>0.92137488461334993</td>
+ <td align="right"> 4.68E-12</td>
+ </tr>
+ <tr>
+ <td>1.765</td>
+ <td>0.9225595178</td>
+ <td>0.92255951786293755</td>
+ <td align="right"> 4.88E-11</td>
+ </tr>
+ <tr>
+ <td>1.770</td>
+ <td>0.9237631277</td>
+ <td>0.9237631277361581</td>
+ <td align="right"> 2.96E-11</td>
+ </tr>
+ <tr>
+ <td>1.775</td>
+ <td>0.9249857211</td>
+ <td>0.92498572112510025</td>
+ <td align="right"> 2.89E-11</td>
+ </tr>
+ <tr>
+ <td>1.780</td>
+ <td>0.9262273062</td>
+ <td>0.92622730619879157</td>
+ <td align="right"> 8.37E-12</td>
+ </tr>
+ <tr>
+ <td>1.785</td>
+ <td>0.9274878926</td>
+ <td>0.92748789239555507</td>
+ <td align="right">-1.97E-10</td>
+ </tr>
+ <tr>
+ <td>1.790</td>
+ <td>0.9287674904</td>
+ <td>0.92876749040057904</td>
+ <td align="right">-3.84E-12</td>
+ </tr>
+ <tr>
+ <td>1.795</td>
+ <td>0.9300661123</td>
+ <td>0.93006611219852275</td>
+ <td align="right">-1.13E-10</td>
+ </tr>
+ <tr>
+ <td>1.800</td>
+ <td>0.9313837710</td>
+ <td>0.93138377097715253</td>
+ <td align="right">-2.97E-11</td>
+ </tr>
+ <tr>
+ <td>1.805</td>
+ <td>0.9327204811</td>
+ <td>0.93272048117993289</td>
+ <td align="right"> 8.20E-11</td>
+ </tr>
+ <tr>
+ <td>1.810</td>
+ <td>0.9340762585</td>
+ <td>0.93407625848467779</td>
+ <td align="right">-2.05E-11</td>
+ </tr>
+ <tr>
+ <td>1.815</td>
+ <td>0.9354511198</td>
+ <td>0.93545111979719375</td>
+ <td align="right"> 8.27E-12</td>
+ </tr>
+ <tr>
+ <td>1.820</td>
+ <td>0.9368450832</td>
+ <td>0.93684508324512517</td>
+ <td align="right"> 4.80E-11</td>
+ </tr>
+ <tr>
+ <td>1.825</td>
+ <td>0.9382581682</td>
+ <td>0.93825816817200214</td>
+ <td align="right">-2.82E-11</td>
+ </tr>
+ <tr>
+ <td>1.830</td>
+ <td>0.9396903951</td>
+ <td>0.93969039513148056</td>
+ <td align="right"> 1.86E-11</td>
+ </tr>
+ <tr>
+ <td>1.835</td>
+ <td>0.9411417859</td>
+ <td>0.94114178588178177</td>
+ <td align="right">-2.64E-11</td>
+ </tr>
+ <tr>
+ <td>1.840</td>
+ <td>0.9426123634</td>
+ <td>0.94261236338031951</td>
+ <td align="right">-2.35E-11</td>
+ </tr>
+ <tr>
+ <td>1.845</td>
+ <td>0.9441021519</td>
+ <td>0.94410215177851575</td>
+ <td align="right">-1.22E-10</td>
+ </tr>
+ <tr>
+ <td>1.850</td>
+ <td>0.9456111764</td>
+ <td>0.94561117639912362</td>
+ <td align="right">-2.02E-12</td>
+ </tr>
+ <tr>
+ <td>1.855</td>
+ <td>0.9471394637</td>
+ <td>0.94713946380190617</td>
+ <td align="right"> 9.43E-11</td>
+ </tr>
+ <tr>
+ <td>1.860</td>
+ <td>0.9486870417</td>
+ <td>0.94868704167359708</td>
+ <td align="right">-2.86E-11</td>
+ </tr>
+ <tr>
+ <td>1.865</td>
+ <td>0.9502539389</td>
+ <td>0.95025393889348797</td>
+ <td align="right">-1.33E-11</td>
+ </tr>
+ <tr>
+ <td>1.870</td>
+ <td>0.9518401855</td>
+ <td>0.95184018551169203</td>
+ <td align="right"> 9.61E-12</td>
+ </tr>
+ <tr>
+ <td>1.875</td>
+ <td>0.9534458127</td>
+ <td>0.95344581274503493</td>
+ <td align="right"> 5.77E-11</td>
+ </tr>
+ <tr>
+ <td>1.880</td>
+ <td>0.9550708530</td>
+ <td>0.95507085297311556</td>
+ <td align="right">-2.73E-11</td>
+ </tr>
+ <tr>
+ <td>1.885</td>
+ <td>0.9567153398</td>
+ <td>0.95671533973453671</td>
+ <td align="right">-6.02E-11</td>
+ </tr>
+ <tr>
+ <td>1.890</td>
+ <td>0.9583793077</td>
+ <td>0.95837930772329927</td>
+ <td align="right"> 1.97E-11</td>
+ </tr>
+ <tr>
+ <td>1.895</td>
+ <td>0.9600627927</td>
+ <td>0.960062792785362</td>
+ <td align="right"> 8.60E-11</td>
+ </tr>
+ <tr>
+ <td>1.900</td>
+ <td>0.9617658319</td>
+ <td>0.96176583191536336</td>
+ <td align="right"> 2.60E-11</td>
+ </tr>
+ <tr>
+ <td>1.905</td>
+ <td>0.9634884632</td>
+ <td>0.96348846325350124</td>
+ <td align="right"> 5.75E-11</td>
+ </tr>
+ <tr>
+ <td>1.910</td>
+ <td>0.9652307261</td>
+ <td>0.96523072608257054</td>
+ <td align="right">-3.05E-11</td>
+ </tr>
+ <tr>
+ <td>1.915</td>
+ <td>0.9669926608</td>
+ <td>0.96699266080453206</td>
+ <td align="right"> 5.78E-13</td>
+ </tr>
+ <tr>
+ <td>1.920</td>
+ <td>0.9687743090</td>
+ <td>0.96877430902013406</td>
+ <td align="right"> 1.66E-11</td>
+ </tr>
+ <tr>
+ <td>1.925</td>
+ <td>0.9705757134</td>
+ <td>0.97057571340334281</td>
+ <td align="right">-3.67E-12</td>
+ </tr>
+ <tr>
+ <td>1.930</td>
+ <td>0.9723969178</td>
+ <td>0.9723969177808085</td>
+ <td align="right">-5.87E-12</td>
+ </tr>
+ <tr>
+ <td>1.935</td>
+ <td>0.9742379672</td>
+ <td>0.97423796710926569</td>
+ <td align="right">-8.59E-11</td>
+ </tr>
+ <tr>
+ <td>1.940</td>
+ <td>0.9760989075</td>
+ <td>0.97609890747347727</td>
+ <td align="right">-2.67E-11</td>
+ </tr>
+ <tr>
+ <td>1.945</td>
+ <td>0.9779797861</td>
+ <td>0.97797978608432246</td>
+ <td align="right">-2.76E-11</td>
+ </tr>
+ <tr>
+ <td>1.950</td>
+ <td>0.9798806513</td>
+ <td>0.9798806512770295</td>
+ <td align="right">-3.65E-11</td>
+ </tr>
+ <tr>
+ <td>1.955</td>
+ <td>0.9818015524</td>
+ <td>0.98180155250954815</td>
+ <td align="right"> 1.02E-10</td>
+ </tr>
+ <tr>
+ <td>1.960</td>
+ <td>0.9837425404</td>
+ <td>0.98374254036106346</td>
+ <td align="right">-5.01E-11</td>
+ </tr>
+ <tr>
+ <td>1.965</td>
+ <td>0.9857036664</td>
+ <td>0.985703666530647</td>
+ <td align="right"> 1.27E-10</td>
+ </tr>
+ <tr>
+ <td>1.970</td>
+ <td>0.9876849838</td>
+ <td>0.98768498383604675</td>
+ <td align="right"> 4.68E-11</td>
+ </tr>
+ <tr>
+ <td>1.975</td>
+ <td>0.9896865462</td>
+ <td>0.98968654618919183</td>
+ <td align="right">-1.77E-11</td>
+ </tr>
+ <tr>
+ <td>1.980</td>
+ <td>0.9917084087</td>
+ <td>0.99170840868869103</td>
+ <td align="right">-3.22E-12</td>
+ </tr>
+ <tr>
+ <td>1.985</td>
+ <td>0.9937506274</td>
+ <td>0.9937506274792185</td>
+ <td align="right"> 6.46E-11</td>
+ </tr>
+ <tr>
+ <td>1.990</td>
+ <td>0.9958132598</td>
+ <td>0.99581325984380575</td>
+ <td align="right"> 4.71E-11</td>
+ </tr>
+ <tr>
+ <td>1.995</td>
+ <td>0.9978963643</td>
+ <td>0.99789636418011041</td>
+ <td align="right">-1.27E-10</td>
+ </tr>
+</table>
+
+
+<h4>The Psi Function</h4>
+
+This table was constructed from the published values in the
+Handbook of Mathematical Functions, by Milton Abramowitz
+and Irene A. Stegun, by Dover (1965), pp 267-270.
+
+Axiom implements the polygamma function which allows for multiple
+derivatives. The Psi function is a special case of the polygamma
+function for zero derivatives. For the purpose of this table it
+is defined as:
+<pre>
+ Psi(x) == polygamma(0,x)
+</pre>
+
+The first column is the point where the Gamma function is evaluated.
+The second column is the value reported in the Handbook.
+The third column is the actual value computed by Axiom at the given point.
+The fourth column is the difference of Axiom's value and the Handbook value.
+
+<table border="1">
+ <tr>
+ <th>point</th>
+ <th>Handbook Value</th>
+ <th>Axiom Computed Value</th>
+ <th>Difference</th>
+ </tr>
+ <tr>
+ <td>1.000</td>
+ <td>-0.5772156649</td>
+ <td>-0.57721566490153275</td>
+ <td align="right">-1.53E-12</td>
+ </tr>
+ <tr>
+ <td>1.005</td>
+ <td>-0.5690209113</td>
+ <td>-0.56902091134438304</td>
+ <td align="right"> -4.43E-11</td>
+ </tr>
+ <tr>
+ <td>1.010</td>
+ <td>-0.5608854579</td>
+ <td>-0.56088545786867472</td>
+ <td align="right"> 3.13E-11</td>
+ </tr>
+ <tr>
+ <td>1.015</td>
+ <td>-0.5528085156</td>
+ <td>-0.55280851559434629</td>
+ <td align="right"> 5.65E-12</td>
+ </tr>
+ <tr>
+ <td>1.020</td>
+ <td>-0.5447893105</td>
+ <td>-0.54478931045617984</td>
+ <td align="right"> 4.38E-11</td>
+ </tr>
+ <tr>
+ <td>1.025</td>
+ <td>-0.5368270828</td>
+ <td>-0.53682708284938863</td>
+ <td align="right"> -4.93E-11</td>
+ </tr>
+ <tr>
+ <td>1.030</td>
+ <td>-0.5289210873</td>
+ <td>-0.5289210872854303</td>
+ <td align="right"> 1.45E-11</td>
+ </tr>
+ <tr>
+ <td>1.035</td>
+ <td>-0.5210705921</td>
+ <td>-0.52107059205771</td>
+ <td align="right"> 4.22E-11</td>
+ </tr>
+ <tr>
+ <td>1.040</td>
+ <td>-0.5132748789</td>
+ <td>-0.51327487891683021</td>
+ <td align="right"> -1.68E-11</td>
+ </tr>
+ <tr>
+ <td>1.045</td>
+ <td>-0.5055332428</td>
+ <td>-0.50553324275508449</td>
+ <td align="right"> 4.49E-11</td>
+ </tr>
+ <tr>
+ <td>1.050</td>
+ <td>-0.4978449913</td>
+ <td>-0.49784499129987031</td>
+ <td align="right"> 1.29E-13</td>
+ </tr>
+ <tr>
+ <td>1.055</td>
+ <td>-0.4902094448</td>
+ <td>-0.49020944481574569</td>
+ <td align="right"> -1.57E-11</td>
+ </tr>
+ <tr>
+ <td>1.060</td>
+ <td>-0.4826259358</td>
+ <td>-0.48262593581482538</td>
+ <td align="right"> -1.48E-11</td>
+ </tr>
+ <tr>
+ <td>1.065</td>
+ <td>-0.4750938088</td>
+ <td>-0.47509380877526647</td>
+ <td align="right"> 2.47E-11</td>
+ </tr>
+ <tr>
+ <td>1.070</td>
+ <td>-0.4676124199</td>
+ <td>-0.46761241986755342</td>
+ <td align="right"> 3.24E-11</td>
+ </tr>
+ <tr>
+ <td>1.075</td>
+ <td>-0.4601811367</td>
+ <td>-0.4601811366883593</td>
+ <td align="right"> 1.16E-11</td>
+ </tr>
+ <tr>
+ <td>1.080</td>
+ <td>-0.4527993380</td>
+ <td>-0.45279933800171246</td>
+ <td align="right"> -1.71E-12</td>
+ </tr>
+ <tr>
+ <td>1.085</td>
+ <td>-0.4454664135</td>
+ <td>-0.44546641348725191</td>
+ <td align="right"> 1.27E-11</td>
+ </tr>
+ <tr>
+ <td>1.090</td>
+ <td>-0.4381817635</td>
+ <td>-0.43818176349533489</td>
+ <td align="right"> 4.66E-12</td>
+ </tr>
+ <tr>
+ <td>1.095</td>
+ <td>-0.4309447988</td>
+ <td>-0.43094479880878706</td>
+ <td align="right"> -8.78E-12</td>
+ </tr>
+ <tr>
+ <td>1.100</td>
+ <td>-0.4237549404</td>
+ <td>-0.42375494041107653</td>
+ <td align="right"> -1.10E-11</td>
+ </tr>
+ <tr>
+ <td>1.105</td>
+ <td>-0.4166116193</td>
+ <td>-0.41661161926071655</td>
+ <td align="right"> 3.92E-11</td>
+ </tr>
+ <tr>
+ <td>1.110</td>
+ <td>-0.4095142761</td>
+ <td>-0.40951427607169383</td>
+ <td align="right"> 2.83E-11</td>
+ </tr>
+ <tr>
+ <td>1.115</td>
+ <td>-0.4024623611</td>
+ <td>-0.40246236109974648</td>
+ <td align="right"> 2.53E-13</td>
+ </tr>
+ <tr>
+ <td>1.120</td>
+ <td>-0.3954553339</td>
+ <td>-0.39545533393429283</td>
+ <td align="right"> -3.42E-11</td>
+ </tr>
+ <tr>
+ <td>1.125</td>
+ <td>-0.3884926633</td>
+ <td>-0.38849266329585463</td>
+ <td align="right"> 4.14E-12</td>
+ </tr>
+ <tr>
+ <td>1.130</td>
+ <td>-0.3815738268</td>
+ <td>-0.38157382683879215</td>
+ <td align="right"> -3.87E-11</td>
+ </tr>
+ <tr>
+ <td>1.135</td>
+ <td>-0.3746983110</td>
+ <td>-0.37469831095919082</td>
+ <td align="right"> 4.08E-11</td>
+ </tr>
+ <tr>
+ <td>1.140</td>
+ <td>-0.3678656106</td>
+ <td>-0.36786561060774969</td>
+ <td align="right"> -7.74E-12</td>
+ </tr>
+ <tr>
+ <td>1.145</td>
+ <td>-0.3610752291</td>
+ <td>-0.361075229107509</td>
+ <td align="right"> -7.50E-12</td>
+ </tr>
+ <tr>
+ <td>1.150</td>
+ <td>-0.3543266780</td>
+ <td>-0.35432667797627904</td>
+ <td align="right"> 2.37E-11</td>
+ </tr>
+ <tr>
+ <td>1.155</td>
+ <td>-0.3476194768</td>
+ <td>-0.34761947675362337</td>
+ <td align="right"> 4.63E-11</td>
+ </tr>
+ <tr>
+ <td>1.160</td>
+ <td>-0.3409531528</td>
+ <td>-0.34095315283226135</td>
+ <td align="right"> -3.22E-11</td>
+ </tr>
+ <tr>
+ <td>1.165</td>
+ <td>-0.3343272413</td>
+ <td>-0.3343272412937619</td>
+ <td align="right"> 6.23E-12</td>
+ </tr>
+ <tr>
+ <td>1.170</td>
+ <td>-0.3277412847</td>
+ <td>-0.3277412847483927</td>
+ <td align="right"> -4.83E-11</td>
+ </tr>
+ <tr>
+ <td>1.175</td>
+ <td>-0.3211948332</td>
+ <td>-0.3211948331790081</td>
+ <td align="right"> 2.09E-11</td>
+ </tr>
+ <tr>
+ <td>1.180</td>
+ <td>-0.3146874438</td>
+ <td>-0.31468744378886082</td>
+ <td align="right"> 1.11E-11</td>
+ </tr>
+ <tr>
+ <td>1.185</td>
+ <td>-0.3082186809</td>
+ <td>-0.30821868085320625</td>
+ <td align="right"> 4.67E-11</td>
+ </tr>
+ <tr>
+ <td>1.190</td>
+ <td>-0.3017881156</td>
+ <td>-0.30178811557461016</td>
+ <td align="right"> 2.53E-11</td>
+ </tr>
+ <tr>
+ <td>1.195</td>
+ <td>-0.2953953259</td>
+ <td>-0.2953953259418296</td>
+ <td align="right"> -4.18E-11</td>
+ </tr>
+ <tr>
+ <td>1.200</td>
+ <td>-0.2890398966</td>
+ <td>-0.28903989659218843</td>
+ <td align="right"> 7.81E-12</td>
+ </tr>
+ <tr>
+ <td>1.205</td>
+ <td>-0.2827214187</td>
+ <td>-0.28272141867731704</td>
+ <td align="right"> 2.26E-11</td>
+ </tr>
+ <tr>
+ <td>1.210</td>
+ <td>-0.2764394897</td>
+ <td>-0.2764394897321919</td>
+ <td align="right"> -3.21E-11</td>
+ </tr>
+ <tr>
+ <td>1.215</td>
+ <td>-0.2701937135</td>
+ <td>-0.27019371354735244</td>
+ <td align="right"> -4.73E-11</td>
+ </tr>
+ <tr>
+ <td>1.220</td>
+ <td>-0.2639837000</td>
+ <td>-0.26398370004422023</td>
+ <td align="right"> -4.42E-11</td>
+ </tr>
+ <tr>
+ <td>1.225</td>
+ <td>-0.2578090652</td>
+ <td>-0.25780906515343338</td>
+ <td align="right"> 4.65E-11</td>
+ </tr>
+ <tr>
+ <td>1.230</td>
+ <td>-0.2516694307</td>
+ <td>-0.25166943069609982</td>
+ <td align="right"> 3.90E-12</td>
+ </tr>
+ <tr>
+ <td>1.235</td>
+ <td>-0.2455644243</td>
+ <td>-0.24556442426789726</td>
+ <td align="right"> 3.21E-11</td>
+ </tr>
+ <tr>
+ <td>1.240</td>
+ <td>-0.2394936791</td>
+ <td>-0.23949367912593666</td>
+ <td align="right"> -2.59E-11</td>
+ </tr>
+ <tr>
+ <td>1.245</td>
+ <td>-0.2334568341</td>
+ <td>-0.23345683407831253</td>
+ <td align="right"> 2.16E-11</td>
+ </tr>
+ <tr>
+ <td>1.250</td>
+ <td>-0.2274535334</td>
+ <td>-0.22745353337626528</td>
+ <td align="right"> 2.37E-11</td>
+ </tr>
+ <tr>
+ <td>1.255</td>
+ <td>-0.2214834266</td>
+ <td>-0.22148342660888165</td>
+ <td align="right"> -8.88E-12</td>
+ </tr>
+ <tr>
+ <td>1.260</td>
+ <td>-0.2155461686</td>
+ <td>-0.21554616860026521</td>
+ <td align="right"> -2.65E-13</td>
+ </tr>
+ <tr>
+ <td>1.265</td>
+ <td>-0.2096414193</td>
+ <td>-0.20964141930911384</td>
+ <td align="right"> -9.11E-12</td>
+ </tr>
+ <tr>
+ <td>1.270</td>
+ <td>-0.2037688437</td>
+ <td>-0.20376884373062343</td>
+ <td align="right"> -3.06E-11</td>
+ </tr>
+ <tr>
+ <td>1.275</td>
+ <td>-0.1979281118</td>
+ <td>-0.19792811180067393</td>
+ <td align="right"> -6.73E-13</td>
+ </tr>
+ <tr>
+ <td>1.280</td>
+ <td>-0.1921188983</td>
+ <td>-0.19211889830222173</td>
+ <td align="right"> -2.22E-12</td>
+ </tr>
+ <tr>
+ <td>1.285</td>
+ <td>-0.1863408828</td>
+ <td>-0.18634088277384209</td>
+ <td align="right"> 2.61E-11</td>
+ </tr>
+ <tr>
+ <td>1.290</td>
+ <td>-0.1805937494</td>
+ <td>-0.1805937494203691</td>
+ <td align="right"> -2.03E-11</td>
+ </tr>
+ <tr>
+ <td>1.295</td>
+ <td>-0.1748771870</td>
+ <td>-0.17487718702556942</td>
+ <td align="right"> -2.55E-11</td>
+ </tr>
+ <tr>
+ <td>1.300</td>
+ <td>-0.1691908889</td>
+ <td>-0.16919088886679934</td>
+ <td align="right"> 3.32E-11</td>
+ </tr>
+ <tr>
+ <td>1.305</td>
+ <td>-0.1635345526</td>
+ <td>-0.163534552631597</td>
+ <td align="right"> -3.15E-11</td>
+ </tr>
+ <tr>
+ <td>1.310</td>
+ <td>-0.1579078803</td>
+ <td>-0.15790788033614178</td>
+ <td align="right"> -3.61E-11</td>
+ </tr>
+ <tr>
+ <td>1.315</td>
+ <td>-0.1523105782</td>
+ <td>-0.15231057824555994</td>
+ <td align="right"> -4.55E-11</td>
+ </tr>
+ <tr>
+ <td>1.320</td>
+ <td>-0.1467423568</td>
+ <td>-0.1467423567959959</td>
+ <td align="right"> 4.00E-12</td>
+ </tr>
+ <tr>
+ <td>1.325</td>
+ <td>-0.1412029305</td>
+ <td>-0.14120293051842803</td>
+ <td align="right"> -1.84E-11</td>
+ </tr>
+ <tr>
+ <td>1.330</td>
+ <td>-0.1356920180</td>
+ <td>-0.13569201796416941</td>
+ <td align="right"> 3.58E-11</td>
+ </tr>
+ <tr>
+ <td>1.335</td>
+ <td>-0.1302093416</td>
+ <td>-0.13020934163201769</td>
+ <td align="right"> -3.20E-11</td>
+ </tr>
+ <tr>
+ <td>1.340</td>
+ <td>-0.1247546279</td>
+ <td>-0.12475462789700376</td>
+ <td align="right"> 2.99E-12</td>
+ </tr>
+ <tr>
+ <td>1.345</td>
+ <td>-0.1193276069</td>
+ <td>-0.11932760694070754</td>
+ <td align="right"> -4.07E-11</td>
+ </tr>
+ <tr>
+ <td>1.350</td>
+ <td>-0.1139280127</td>
+ <td>-0.11392801268308839</td>
+ <td align="right"> 1.69E-11</td>
+ </tr>
+ <tr>
+ <td>1.355</td>
+ <td>-0.1085555827</td>
+ <td>-0.10855558271580501</td>
+ <td align="right"> -1.58E-11</td>
+ </tr>
+ <tr>
+ <td>1.360</td>
+ <td>-0.1032100582</td>
+ <td>-0.10321005823697738</td>
+ <td align="right"> -3.69E-11</td>
+ </tr>
+ <tr>
+ <td>1.365</td>
+ <td>-0.0978911840</td>
+ <td>-0.097891183987354968</td>
+ <td align="right"> 1.26E-11</td>
+ </tr>
+ <tr>
+ <td>1.370</td>
+ <td>-0.0925987082</td>
+ <td>-0.092598708187860979</td>
+ <td align="right"> 1.21E-11</td>
+ </tr>
+ <tr>
+ <td>1.375</td>
+ <td>-0.0873323825</td>
+ <td>-0.087332382478473081</td>
+ <td align="right"> 2.15E-11</td>
+ </tr>
+ <tr>
+ <td>1.380</td>
+ <td>-0.0820919619</td>
+ <td>-0.082091961858406615</td>
+ <td align="right"> 4.15E-11</td>
+ </tr>
+ <tr>
+ <td>1.385</td>
+ <td>-0.0768772046</td>
+ <td>-0.076877204627574525</td>
+ <td align="right"> -2.75E-11</td>
+ </tr>
+ <tr>
+ <td>1.390</td>
+ <td>-0.0716878723</td>
+ <td>-0.071687872329281643</td>
+ <td align="right"> -2.92E-11</td>
+ </tr>
+ <tr>
+ <td>1.395</td>
+ <td>-0.0665237297</td>
+ <td>-0.066523729694132228</td>
+ <td align="right"> 5.86E-12</td>
+ </tr>
+ <tr>
+ <td>1.400</td>
+ <td>-0.0613845446</td>
+ <td>-0.061384544585116108</td>
+ <td align="right"> 1.48E-11</td>
+ </tr>
+ <tr>
+ <td>1.405</td>
+ <td>-0.0562700879</td>
+ <td>-0.056270087943841696</td>
+ <td align="right"> -4.38E-11</td>
+ </tr>
+ <tr>
+ <td>1.410</td>
+ <td>-0.0511801337</td>
+ <td>-0.051180133737897426</td>
+ <td align="right"> -3.78E-11</td>
+ </tr>
+ <tr>
+ <td>1.415</td>
+ <td>-0.0461144589</td>
+ <td>-0.04.6114458909301992</td>
+ <td align="right"> -9.30E-12</td>
+ </tr>
+ <tr>
+ <td>1.420</td>
+ <td>-0.0410728433</td>
+ <td>-0.041072843324024277</td>
+ <td align="right"> -2.40E-11</td>
+ </tr>
+ <tr>
+ <td>1.425</td>
+ <td>-0.0360550697</td>
+ <td>-0.036055069722547906</td>
+ <td align="right"> -2.25E-11</td>
+ </tr>
+ <tr>
+ <td>1.430</td>
+ <td>-0.0310609237</td>
+ <td>-0.031060923671447194</td>
+ <td align="right"> 2.85E-11</td>
+ </tr>
+ <tr>
+ <td>1.435</td>
+ <td>-0.0260901935</td>
+ <td>-0.02609019351596098</td>
+ <td align="right"> -1.59E-11</td>
+ </tr>
+ <tr>
+ <td>1.440</td>
+ <td>-0.0211426703</td>
+ <td>-0.021142670333530678</td>
+ <td align="right"> -3.35E-11</td>
+ </tr>
+ <tr>
+ <td>1.445</td>
+ <td>-0.0162181479</td>
+ <td>-0.016218147888283685</td>
+ <td align="right"> 1.17E-11</td>
+ </tr>
+ <tr>
+ <td>1.450</td>
+ <td>-0.0113164226</td>
+ <td>-0.011316422586445718</td>
+ <td align="right"> 1.35E-11</td>
+ </tr>
+ <tr>
+ <td>1.455</td>
+ <td>-0.0064372934</td>
+ <td>-0.0064372934326406561</td>
+ <td align="right"> -3.26E-11</td>
+ </tr>
+ <tr>
+ <td>1.460</td>
+ <td>-0.0015805620</td>
+ <td>-0.0015805619870833398</td>
+ <td align="right"> 1.29E-11</td>
+ </tr>
+ <tr>
+ <td>1.465</td>
+ <td>0.0032539677</td>
+ <td>0.0032539676763745362</td>
+ <td align="right"> -2.36E-11</td>
+ </tr>
+ <tr>
+ <td>1.470</td>
+ <td>0.0080664890</td>
+ <td>0.0080664890113649745</td>
+ <td align="right"> 1.13E-11</td>
+ </tr>
+ <tr>
+ <td>1.475</td>
+ <td>0.0128571930</td>
+ <td>0.012857193039295334</td>
+ <td align="right"> 3.92E-11</td>
+ </tr>
+ <tr>
+ <td>1.480</td>
+ <td>0.0176262684</td>
+ <td>0.017626268388849287</td>
+ <td align="right"> -1.11E-11</td>
+ </tr>
+ <tr>
+ <td>1.485</td>
+ <td>0.0223739013</td>
+ <td>0.022373901334705404</td>
+ <td align="right"> 3.47E-11</td>
+ </tr>
+ <tr>
+ <td>1.490</td>
+ <td>0.0271002758</td>
+ <td>0.027100275835486465</td>
+ <td align="right"> 3.54E-11</td>
+ </tr>
+ <tr>
+ <td>1.495</td>
+ <td>0.0318055736</td>
+ <td>0.031805573570971468</td>
+ <td align="right"> -2.90E-11</td>
+ </tr>
+ <tr>
+ <td>1.500</td>
+ <td>0.0364899740</td>
+ <td>0.036489973978576673</td>
+ <td align="right"> -2.14E-11</td>
+ </tr>
+ <tr>
+ <td>1.505</td>
+ <td>0.0411536543</td>
+ <td>0.041153654289123542</td>
+ <td align="right"> -1.08E-11</td>
+ </tr>
+ <tr>
+ <td>1.510</td>
+ <td>0.0457967896</td>
+ <td>0.045796789561914686</td>
+ <td align="right"> -3.80E-11</td>
+ </tr>
+ <tr>
+ <td>1.515</td>
+ <td>0.0504195527</td>
+ <td>0.050419552719128236</td>
+ <td align="right"> 1.91E-11</td>
+ </tr>
+ <tr>
+ <td>1.520</td>
+ <td>0.0550221146</td>
+ <td>0.055022114579551307</td>
+ <td align="right"> -2.04E-11</td>
+ </tr>
+ <tr>
+ <td>1.525</td>
+ <td>0.0596046439</td>
+ <td>0.05960464389166209</td>
+ <td align="right"> -8.33E-12</td>
+ </tr>
+ <tr>
+ <td>1.530</td>
+ <td>0.0641673074</td>
+ <td>0.064167307366077231</td>
+ <td align="right"> -3.39E-11</td>
+ </tr>
+ <tr>
+ <td>1.535</td>
+ <td>0.0687102697</td>
+ <td>0.068710269707385141</td>
+ <td align="right"> 7.38E-12</td>
+ </tr>
+ <tr>
+ <td>1.540</td>
+ <td>0.0732336936</td>
+ <td>0.073233693645366138</td>
+ <td align="right"> 4.53E-11</td>
+ </tr>
+ <tr>
+ <td>1.545</td>
+ <td>0.0777377300</td>
+ <td>0.077737739965624497</td>
+ <td align="right"> 9.96E-9</td>
+ </tr>
+ <tr>
+ <td>1.550</td>
+ <td>0.0822225675</td>
+ <td>0.082222567539644631</td>
+ <td align="right"> 3.96E-11</td>
+ </tr>
+ <tr>
+ <td>1.555</td>
+ <td>0.0866883334</td>
+ <td>0.086688333354268288</td>
+ <td align="right"> -4.57E-11</td>
+ </tr>
+ <tr>
+ <td>1.560</td>
+ <td>0.0911351925</td>
+ <td>0.091135192540635401</td>
+ <td align="right"> 4.06E-11</td>
+ </tr>
+ <tr>
+ <td>1.565</td>
+ <td>0.0955632984</td>
+ <td>0.095563298402570163</td>
+ <td align="right"> 2.57E-12</td>
+ </tr>
+ <tr>
+ <td>1.570</td>
+ <td>0.0999728024</td>
+ <td>0.099972802444444731</td>
+ <td align="right"> 4.44E-11</td>
+ </tr>
+ <tr>
+ <td>1.575</td>
+ <td>0.1043638544</td>
+ <td>0.10436385439851947</td>
+ <td align="right"> -1.48E-12</td>
+ </tr>
+ <tr>
+ <td>1.580</td>
+ <td>0.1087366023</td>
+ <td>0.10873660225178161</td>
+ <td align="right"> -4.82E-11</td>
+ </tr>
+ <tr>
+ <td>1.585</td>
+ <td>0.1130911923</td>
+ <td>0.11309119227228603</td>
+ <td align="right"> -2.77E-11</td>
+ </tr>
+ <tr>
+ <td>1.590</td>
+ <td>0.1174277690</td>
+ <td>0.11742776903501095</td>
+ <td align="right"> 3.50E-11</td>
+ </tr>
+ <tr>
+ <td>1.595</td>
+ <td>0.1217464754</td>
+ <td>0.12174647544723916</td>
+ <td align="right"> 4.72E-11</td>
+ </tr>
+ <tr>
+ <td>1.600</td>
+ <td>0.1260474528</td>
+ <td>0.12604745277347584</td>
+ <td align="right"> -2.65E-11</td>
+ </tr>
+ <tr>
+ <td>1.605</td>
+ <td>0.1303308407</td>
+ <td>0.13033084065991318</td>
+ <td align="right"> -4.00E-11</td>
+ </tr>
+ <tr>
+ <td>1.610</td>
+ <td>0.1345967772</td>
+ <td>0.13459677715844587</td>
+ <td align="right"> -4.15E-11</td>
+ </tr>
+ <tr>
+ <td>1.615</td>
+ <td>0.1388453988</td>
+ <td>0.13884539875025736</td>
+ <td align="right"> -4.97E-11</td>
+ </tr>
+ <tr>
+ <td>1.620</td>
+ <td>0.1430768404</td>
+ <td>0.14307684036898005</td>
+ <td align="right"> -3.10E-11</td>
+ </tr>
+ <tr>
+ <td>1.625</td>
+ <td>0.1472912354</td>
+ <td>0.14729123542343325</td>
+ <td align="right"> 2.34E-11</td>
+ </tr>
+ <tr>
+ <td>1.630</td>
+ <td>0.1514887158</td>
+ <td>0.15148871581995815</td>
+ <td align="right"> 1.99E-11</td>
+ </tr>
+ <tr>
+ <td>1.635</td>
+ <td>0.1556694120</td>
+ <td>0.15566941198435302</td>
+ <td align="right"> -1.56E-11</td>
+ </tr>
+ <tr>
+ <td>1.640</td>
+ <td>0.1598334529</td>
+ <td>0.15983345288341522</td>
+ <td align="right"> -1.65E-11</td>
+ </tr>
+ <tr>
+ <td>1.645</td>
+ <td>0.1639809660</td>
+ <td>0.16398096604610457</td>
+ <td align="right"> 4.61E-11</td>
+ </tr>
+ <tr>
+ <td>1.650</td>
+ <td>0.1681120776</td>
+ <td>0.16811207758432767</td>
+ <td align="right"> -1.56E-11</td>
+ </tr>
+ <tr>
+ <td>1.655</td>
+ <td>0.1722269122</td>
+ <td>0.17222691221335784</td>
+ <td align="right"> 1.33E-11</td>
+ </tr>
+ <tr>
+ <td>1.660</td>
+ <td>0.1763255933</td>
+ <td>0.17632559327189457</td>
+ <td align="right"> -2.81E-11</td>
+ </tr>
+ <tr>
+ <td>1.665</td>
+ <td>0.1804082427</td>
+ <td>0.18040824274177392</td>
+ <td align="right"> 4.17E-11</td>
+ </tr>
+ <tr>
+ <td>1.670</td>
+ <td>0.1844749813</td>
+ <td>0.1844749812673292</td>
+ <td align="right"> -3.26E-11</td>
+ </tr>
+ <tr>
+ <td>1.675</td>
+ <td>0.1885259282</td>
+ <td>0.18852592817442249</td>
+ <td align="right"> -2.55E-11</td>
+ </tr>
+ <tr>
+ <td>1.680</td>
+ <td>0.1925612015</td>
+ <td>0.19256120148913258</td>
+ <td align="right"> -1.08E-11</td>
+ </tr>
+ <tr>
+ <td>1.685</td>
+ <td>0.1965809180</td>
+ <td>0.19658091795613342</td>
+ <td align="right"> -4.38E-11</td>
+ </tr>
+ <tr>
+ <td>1.690</td>
+ <td>0.2005851931</td>
+ <td>0.20058519305674649</td>
+ <td align="right"> -4.32E-11</td>
+ </tr>
+ <tr>
+ <td>1.695</td>
+ <td>0.2045741410</td>
+ <td>0.20457414102668603</td>
+ <td align="right"> 2.66E-11</td>
+ </tr>
+ <tr>
+ <td>1.700</td>
+ <td>0.2085478749</td>
+ <td>0.20854787487349435</td>
+ <td align="right"> -2.65E-11</td>
+ </tr>
+ <tr>
+ <td>1.705</td>
+ <td>0.2125065064</td>
+ <td>0.21250650639368796</td>
+ <td align="right"> -6.31E-12</td>
+ </tr>
+ <tr>
+ <td>1.710</td>
+ <td>0.2164501462</td>
+ <td>0.21645014618960501</td>
+ <td align="right"> -1.03E-11</td>
+ </tr>
+ <tr>
+ <td>1.715</td>
+ <td>0.2203789037</td>
+ <td>0.2203789036859658</td>
+ <td align="right"> -1.40E-11</td>
+ </tr>
+ <tr>
+ <td>1.720</td>
+ <td>0.2242928871</td>
+ <td>0.22429288714615725</td>
+ <td align="right"> 4.61E-11</td>
+ </tr>
+ <tr>
+ <td>1.725</td>
+ <td>0.2281922037</td>
+ <td>0.22819220368823745</td>
+ <td align="right"> -1.17E-11</td>
+ </tr>
+ <tr>
+ <td>1.730</td>
+ <td>0.2320769593</td>
+ <td>0.23207695930067274</td>
+ <td align="right"> 6.72E-13</td>
+ </tr>
+ <tr>
+ <td>1.735</td>
+ <td>0.2359472589</td>
+ <td>0.23594725885781176</td>
+ <td align="right"> -4.21E-11</td>
+ </tr>
+ <tr>
+ <td>1.740</td>
+ <td>0.2398032061</td>
+ <td>0.23980320613509676</td>
+ <td align="right"> 3.50E-11</td>
+ </tr>
+ <tr>
+ <td>1.745</td>
+ <td>0.2436449038</td>
+ <td>0.24364490382402559</td>
+ <td align="right"> 2.40E-11</td>
+ </tr>
+ <tr>
+ <td>1.750</td>
+ <td>0.2474724535</td>
+ <td>0.2474724535468612</td>
+ <td align="right"> 4.68E-11</td>
+ </tr>
+ <tr>
+ <td>1.755</td>
+ <td>0.2512859559</td>
+ <td>0.25128595587109781</td>
+ <td align="right"> -2.89E-11</td>
+ </tr>
+ <tr>
+ <td>1.760</td>
+ <td>0.2550855103</td>
+ <td>0.25508551032368809</td>
+ <td align="right"> 2.36E-11</td>
+ </tr>
+ <tr>
+ <td>1.765</td>
+ <td>0.2588712154</td>
+ <td>0.25887121540503744</td>
+ <td align="right"> 5.03E-12</td>
+ </tr>
+ <tr>
+ <td>1.770</td>
+ <td>0.2626431686</td>
+ <td>0.26264316860276249</td>
+ <td align="right"> 2.76E-12</td>
+ </tr>
+ <tr>
+ <td>1.775</td>
+ <td>0.2664014664</td>
+ <td>0.2664014664052331</td>
+ <td align="right"> 5.23E-12</td>
+ </tr>
+ <tr>
+ <td>1.780</td>
+ <td>0.2701462043</td>
+ <td>0.27014620431488368</td>
+ <td align="right"> 1.48E-11</td>
+ </tr>
+ <tr>
+ <td>1.785</td>
+ <td>0.2738774769</td>
+ <td>0.27387747686131236</td>
+ <td align="right"> -3.86E-11</td>
+ </tr>
+ <tr>
+ <td>1.790</td>
+ <td>0.2775953776</td>
+ <td>0.27759537761416786</td>
+ <td align="right"> 1.41E-11</td>
+ </tr>
+ <tr>
+ <td>1.795</td>
+ <td>0.2812999992</td>
+ <td>0.2812999991958266</td>
+ <td align="right"> -4.17E-12</td>
+ </tr>
+ <tr>
+ <td>1.800</td>
+ <td>0.2849914333</td>
+ <td>0.2849914332938619</td>
+ <td align="right"> -6.13E-12</td>
+ </tr>
+ <tr>
+ <td>1.805</td>
+ <td>0.2886697707</td>
+ <td>0.28866977067331689</td>
+ <td align="right"> -2.66E-11</td>
+ </tr>
+ <tr>
+ <td>1.810</td>
+ <td>0.2923351012</td>
+ <td>0.29233510118877948</td>
+ <td align="right"> -1.12E-11</td>
+ </tr>
+ <tr>
+ <td>1.815</td>
+ <td>0.2959875138</td>
+ <td>0.29598751379626109</td>
+ <td align="right"> -3.73E-12</td>
+ </tr>
+ <tr>
+ <td>1.820</td>
+ <td>0.2996270966</td>
+ <td>0.29962709656488773</td>
+ <td align="right"> -3.51E-11</td>
+ </tr>
+ <tr>
+ <td>1.825</td>
+ <td>0.3032539367</td>
+ <td>0.30325393668840539</td>
+ <td align="right"> -1.15E-11</td>
+ </tr>
+ <tr>
+ <td>1.830</td>
+ <td>0.3068681205</td>
+ <td>0.30686812049650136</td>
+ <td align="right"> -3.49E-12</td>
+ </tr>
+ <tr>
+ <td>1.835</td>
+ <td>0.3104697335</td>
+ <td>0.31046973346594764</td>
+ <td align="right"> -3.40E-11</td>
+ </tr>
+ <tr>
+ <td>1.840</td>
+ <td>0.3140588602</td>
+ <td>0.31405886023156859</td>
+ <td align="right"> 3.15E-11</td>
+ </tr>
+ <tr>
+ <td>1.845</td>
+ <td>0.3176355846</td>
+ <td>0.31763558459703256</td>
+ <td align="right"> -2.96E-12</td>
+ </tr>
+ <tr>
+ <td>1.850</td>
+ <td>0.3211999895</td>
+ <td>0.32119998954547946</td>
+ <td align="right"> 4.54E-11</td>
+ </tr>
+ <tr>
+ <td>1.855</td>
+ <td>0.3247521572</td>
+ <td>0.32475215724997797</td>
+ <td align="right"> 4.99E-11</td>
+ </tr>
+ <tr>
+ <td>1.860</td>
+ <td>0.3282921691</td>
+ <td>0.32829216908382075</td>
+ <td align="right"> -1.61E-11</td>
+ </tr>
+ <tr>
+ <td>1.865</td>
+ <td>0.3318201056</td>
+ <td>0.33182010563065989</td>
+ <td align="right"> 3.06E-11</td>
+ </tr>
+ <tr>
+ <td>1.870</td>
+ <td>0.3353360467</td>
+ <td>0.33533604669448569</td>
+ <td align="right"> -5.51E-12</td>
+ </tr>
+ <tr>
+ <td>1.875</td>
+ <td>0.3388400713</td>
+ <td>0.33884007130944738</td>
+ <td align="right"> 9.44E-12</td>
+ </tr>
+ <tr>
+ <td>1.880</td>
+ <td>0.3423322577</td>
+ <td>0.34233225774952925</td>
+ <td align="right"> 4.95E-11</td>
+ </tr>
+ <tr>
+ <td>1.885</td>
+ <td>0.3458126835</td>
+ <td>0.34581268353806771</td>
+ <td align="right"> 3.80E-11</td>
+ </tr>
+ <tr>
+ <td>1.890</td>
+ <td>0.3492814255</td>
+ <td>0.34928142545713492</td>
+ <td align="right"> -4.28E-11</td>
+ </tr>
+ <tr>
+ <td>1.895</td>
+ <td>0.3527385596</td>
+ <td>0.35273855955676792</td>
+ <td align="right"> -4.32E-11</td>
+ </tr>
+ <tr>
+ <td>1.900</td>
+ <td>0.3561841612</td>
+ <td>0.35618416116406026</td>
+ <td align="right"> -3.59E-11</td>
+ </tr>
+ <tr>
+ <td>1.905</td>
+ <td>0.3596183049</td>
+ <td>0.35961830489211799</td>
+ <td align="right"> -7.88E-12</td>
+ </tr>
+ <tr>
+ <td>1.910</td>
+ <td>0.3630410646</td>
+ <td>0.36304106464888108</td>
+ <td align="right"> 4.88E-11</td>
+ </tr>
+ <tr>
+ <td>1.915</td>
+ <td>0.3664525136</td>
+ <td>0.36645251364580167</td>
+ <td align="right"> 4.58E-11</td>
+ </tr>
+ <tr>
+ <td>1.920</td>
+ <td>0.3698527244</td>
+ <td>0.36985272440640171</td>
+ <td align="right"> 6.40E-12</td>
+ </tr>
+ <tr>
+ <td>1.925</td>
+ <td>0.3732417688</td>
+ <td>0.37324176877469795</td>
+ <td align="right"> -2.53E-11</td>
+ </tr>
+ <tr>
+ <td>1.930</td>
+ <td>0.3766197179</td>
+ <td>0.37661971792349891</td>
+ <td align="right"> 2.34E-11</td>
+ </tr>
+ <tr>
+ <td>1.935</td>
+ <td>0.3799866424</td>
+ <td>0.37998664236258128</td>
+ <td align="right"> -3.74E-11</td>
+ </tr>
+ <tr>
+ <td>1.940</td>
+ <td>0.3833426119</td>
+ <td>0.38334261194674013</td>
+ <td align="right"> 4.67E-11</td>
+ </tr>
+ <tr>
+ <td>1.945</td>
+ <td>0.3866876959</td>
+ <td>0.38668769588372298</td>
+ <td align="right"> -1.62E-11</td>
+ </tr>
+ <tr>
+ <td>1.950</td>
+ <td>0.3900219627</td>
+ <td>0.39002196274204304</td>
+ <td align="right"> 4.20E-11</td>
+ </tr>
+ <tr>
+ <td>1.955</td>
+ <td>0.3933454805</td>
+ <td>0.39334548045868012</td>
+ <td align="right"> -4.13E-11</td>
+ </tr>
+ <tr>
+ <td>1.960</td>
+ <td>0.3966583163</td>
+ <td>0.39665831634666171</td>
+ <td align="right"> 4.66E-11</td>
+ </tr>
+ <tr>
+ <td>1.965</td>
+ <td>0.3999605371</td>
+ <td>0.39996053710254509</td>
+ <td align="right"> 2.54E-12</td>
+ </tr>
+ <tr>
+ <td>1.970</td>
+ <td>0.4032522088</td>
+ <td>0.40325220881377177</td>
+ <td align="right"> 1.37E-11</td>
+ </tr>
+ <tr>
+ <td>1.975</td>
+ <td>0.4065333970</td>
+ <td>0.40653339696592627</td>
+ <td align="right"> -3.40E-11</td>
+ </tr>
+ <tr>
+ <td>1.980</td>
+ <td>0.4098041664</td>
+ <td>0.40980416644989071</td>
+ <td align="right"> 4.98E-11</td>
+ </tr>
+ <tr>
+ <td>1.985</td>
+ <td>0.4130645816</td>
+ <td>0.41306458156888626</td>
+ <td align="right"> -3.11E-11</td>
+ </tr>
+ <tr>
+ <td>1.990</td>
+ <td>0.4163147060</td>
+ <td>0.41631470604541487</td>
+ <td align="right"> 4.54E-11</td>
+ </tr>
+ <tr>
+ <td>1.995</td>
+ <td>0.4195546030</td>
+ <td>0.41955460302810832</td>
+ <td align="right"> 2.81E-11</td>
+ </tr>
+ <tr>
+ <td>2.000</td>
+ <td>0.4227843351</td>
+ <td>0.42278433509846725</td>
+ <td align="right"> -1.53E-12</td>
+ </tr>
+</table>
+<<page foot>>
+@
+
\subsection{draw.xhtml}
<<draw.xhtml>>=
<<standard head>>
@@ -21613,6 +42463,10 @@ infinity; the step size is any positive integer.
<td><a href="ocwmit18085.xhtml"><b>Mathematical Methods</b></a></td>
<td>MIT 18-08 Mathematical Methods for Engineers Course Notes</td>
</tr>
+ <tr>
+ <td><a href="cats.xhtml"><b>CATS</b></a></td>
+ <td>Computer Algebra Test Suite</td>
+ </tr>
</table>
<<page foot>>
@
@@ -22148,7 +43002,7 @@ static char axiom_bits[] = {
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00};
@
-
+
\section{License}
<<license>>=
--Copyright (c) 2007 Arthur C. Ralfs
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