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[Axiom-developer] 20080221.01.tpd.patch (7099: complex gamma function in
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From: |
daly |
|
Subject: |
[Axiom-developer] 20080221.01.tpd.patch (7099: complex gamma function investigation) |
|
Date: |
Thu, 21 Feb 2008 00:28:39 -0600 |
fixed 7099: complex Gamma bug
Note that at the value 1.0+4.6i there is a radical departure between
the table and the computed values in the imaginary part of the value
even though the real part is exact.
For the complex number 1+4.5i using Axiom's algorithm we create a
list of 4 values,
[the complex number
the Abramowitz value
the Axiom expression
the difference of (Axiom expression) - (Abramowitz)]
So at the complex value of 1+4.5i we find:
[1. + 4.5 * %i,
- 5.397606238984 + 3.035196999922 * %i,_
log(Gamma(1. + 4.5*%i)),_
log(Gamma(1. + 4.5*%i))-(- 5.397606238984 + 3.035196999922 * %i)]
[1. + 4.5 %i,
- 5.3976062389840003 + 3.0351969999219999 %i,
- 5.3976062389839621 + 3.0351969999216122 %i,
3.8191672047105385E-14 - 3.8768988019910466E-13 %i]
which is correct to about 13 places. We also compute the same values
using the formula 6.1.40 on page 257 in Abramowitz:
[1. + 4.5 * %i,
- 5.397606238984 + 3.035196999922 * %i,
H(1. + 4.5 * %i),
H(1. + 4.5 * %i)-(- 5.397606238984 + 3.035196999922 * %i)],
[1.0 + 4.5 %i,
- 5.397606238984 + 3.035196999922 %i,
- 5.3976062390513914695 + 3.035196999831937613 %i,
- 0.673914695 E -10 - 0.90062387 E -10 %i]
which is correct to about 10 places.
At the complex value of 1+4.6i we find:
[1. + 4.6 * %i,
- 5.543696418304 + 3.187112279389 * %i,_
log(Gamma(1. + 4.6*%i)),_
log(Gamma(1. + 4.6*%i))-(- 5.543696418304 + 3.187112279389 * %i)]
[1. + 4.6 %i,
- 5.5436964183040001 + 3.1871122793889999 %i,
- 5.5436964183041857 - 3.0960730277910646 %i,
- 1.8562928971732617E-13 - 6.2831853071800641 %i]
which is wildly different from Abramowitz in the imaginary part.
But using the 6.1.40 formula we compute:
[1. + 4.6 * %i,
- 5.543696418304 + 3.187112279389 * %i,
H(1. + 4.6 * %i),
H(1. + 4.6 * %i)-(- 5.543696418304 + 3.187112279389 * %i)],
[1.0 + 4.6 %i,
- 5.543696418304 + 3.187112279389 %i,
- 5.543696418361390699 + 3.1871122793208820187 %i,
- 0.57390699 E -10 - 0.681179813 E -10 %i]
which is again correct to about 10 places.
The Abramowitz formula is computed in src/input/gamma.input.pamphlet
==================================================================
diff --git a/changelog b/changelog
index 90b2e1c..c7165ea 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080221 tpd src/input/gamma.input investigate complex gamma issues
20080220 tpd src/hyper/bookvol11 add additional hyperdoc page translations
20080219 tpd src/hyper/Makefile handle hyperdoc bitmaps properly
20080219 tpd src/Makefile handle hyperdoc bitmaps properly
diff --git a/src/input/gamma.input.pamphlet b/src/input/gamma.input.pamphlet
index 6b5f38b..a8c533c 100644
--- a/src/input/gamma.input.pamphlet
+++ b/src/input/gamma.input.pamphlet
@@ -23,7 +23,10 @@ Dover Publications, Inc. New York 1965. pp267-270
)set message auto off
)clear all
---S 1 of 4
+@
+\section{Gamma 1.000 to 1.995 by 0.005}
+<<*>>=
+--S 1 of 10
[[1.000,1.0000000000,Gamma(1.000),Gamma(1.000)-1.0000000000],_
[1.005,0.9971385354,Gamma(1.005),Gamma(1.005)-0.9971385354],_
[1.010,0.9943258512,Gamma(1.010),Gamma(1.010)-0.9943258512],_
@@ -756,7 +759,8 @@ Dover Publications, Inc. New York 1965. pp267-270
--R Type: List List
DoubleFloat
--E 1
---S 2 of 4
+\section{Psi}
+--S 2 of 10
Psi(x:DFLOAT):DFLOAT==polygamma(0,x)
--R
--R Function declaration Psi : DoubleFloat -> DoubleFloat has been added
@@ -770,9 +774,10 @@ second column is the reference value of Psi from the book
Abramowitz and Stegun, ``Handbook of Mathematical Functions'',
Dover Publications, Inc. New York 1965. pp267-270
+\section{Psi 1.000 to 2.000 by 0.005}
<<*>>=
---S 3 of 4
+--S 3 of 10
[[1.000, -0.5772156649, Psi(1.000), Psi(1.000)- -0.5772156649],_
[1.005, -0.5690209113, Psi(1.005), Psi(1.005)- -0.5690209113],_
[1.010, -0.5608854579, Psi(1.010), Psi(1.010)- -0.5608854579],_
@@ -1535,6 +1540,7 @@ Dover Publications, Inc. New York 1965. pp267-270
--R Type: List List
DoubleFloat
--E 3
@
+\section{log(Gamma) 1+0.0i to 1+10.0i by 0+0.1i}
In the following table there are 4 columns. The first column
is the argument of complex Gamma, ranging from 1.0 + 0i to 1.0+10i
second column is the reference value of complex Gamma from the book
@@ -1545,7 +1551,7 @@ Note that at the value 1.0+4.6i there is a radical
departure between
the table and the computed values in the imaginary part of the value
even though the real part is exact.
<<*>>=
---S 4 of 4
+--S 4 of 10
[[1. + 0.0 * %i,0.,log(Gamma(1. + 0.0 * %i)),log(Gamma(1. + 0.0 * %i))-0.0],_
[1. + 0.1 * %i, -0.008197780565 - 0.057322940417 * %i,_
log(Gamma(1. + 0.1 * %i)),_
@@ -1854,8 +1860,8 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R
--R [1. + 0.10000000000000001 %i,
--R - 8.1977805649999999E-3 - 5.7322940417E-2 %i,
---R - 8.1977805654060397E-3 - 5.7322940416719668E-2 %i,
---R - 4.0603978512798733E-13 + 2.8033131371785203E-13 %i]
+--R - 8.1977805654051359E-3 - 5.7322940416719675E-2 %i,
+--R - 4.0513599419700341E-13 + 2.8032437482394812E-13 %i]
--R ,
--R
--R [1. + 0.20000000000000001 %i, - 3.2476292317999998E-2 - 0.112302222644
%i,
@@ -1964,14 +1970,14 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R
--R [1. + 2.2000000000000002 %i,
--R - 2.1425842092959999 + 0.28184565842600001 %i,
---R - 2.1425842092962606 + 0.28184565842564124 %i,
---R - 2.6068036618198676E-13 - 3.5876857040761934E-13 %i]
+--R - 2.1425842092962588 + 0.28184565842564124 %i,
+--R - 2.5890400934258651E-13 - 3.5876857040761934E-13 %i]
--R ,
--R
--R [1. + 2.2999999999999998 %i,
--R - 2.2774381922039999 + 0.36461404894999999 %i,
---R - 2.2774381922042561 + 0.36461404895017457 %i,
---R - 2.5623947408348613E-13 + 1.7458257062230587E-13 %i]
+--R - 2.2774381922042544 + 0.36461404895017457 %i,
+--R - 2.5446311724408588E-13 + 1.7458257062230587E-13 %i]
--R ,
--R
--R [1. + 2.3999999999999999 %i,
@@ -1981,8 +1987,8 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R ,
--R
--R [1. + 2.5 %i, - 2.549906842495 + 0.54260440585199998 %i,
---R - 2.5499068424946216 + 0.54260440585243641 %i,
---R 3.7836400679225335E-13 + 4.3642867098014904E-13 %i]
+--R - 2.5499068424946199 + 0.54260440585243641 %i,
+--R 3.801403636316536E-13 + 4.3642867098014904E-13 %i]
--R ,
--R
--R [1. + 2.6000000000000001 %i,
@@ -2029,8 +2035,8 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R ,
--R
--R [1. + 3.3999999999999999 %i, - 3.809881261823 + 1.5216522746729999 %i,
---R - 3.8098812618231577 + 1.5216522746726655 %i,
---R - 1.5765166949677223E-13 - 3.3439917501709715E-13 %i]
+--R - 3.8098812618231559 + 1.5216522746726655 %i,
+--R - 1.5587531265737198E-13 - 3.3439917501709715E-13 %i]
--R ,
--R
--R [1. + 3.5 %i, - 3.9524671261890001 + 1.6461926242689999 %i,
@@ -2154,8 +2160,8 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R ,
--R
--R [1. + 5.9000000000000004 %i, - 7.4612836194290004 + 5.3434791013530001
%i,
---R - 7.4612836194293823 - 0.93970620582698861 %i,
---R - 3.8191672047105385E-13 - 6.2831853071799886 %i]
+--R - 7.4612836194293806 - 0.93970620582698861 %i,
+--R - 3.801403636316536E-13 - 6.2831853071799886 %i]
--R ,
--R
--R [1. + 6. %i, - 7.6099596929509996 + 5.5220531255149998 %i,
@@ -2214,8 +2220,8 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R ,
--R
--R [1. + 7.0999999999999996 %i, - 9.2536679950150003 + 7.5903262351840004
%i,
---R - 9.2536679950154568 + 1.3071409280042976 %i,
---R - 4.5652370772586437E-13 - 6.2831853071797026 %i]
+--R - 9.253667995015455 + 1.3071409280042976 %i,
+--R - 4.5474735088646412E-13 - 6.2831853071797026 %i]
--R ,
--R
--R [1. + 7.2000000000000002 %i, - 9.4037545067079993 + 7.7871999928770004
%i,
@@ -2234,8 +2240,8 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R ,
--R
--R [1. + 7.5 %i, - 9.8545824074859993 + 8.3860530880889996 %i,
---R - 9.8545824074859194 + 2.1028677809095977 %i,
---R 7.9936057773011271E-14 - 6.2831853071794015 %i]
+--R - 9.8545824074859176 + 2.1028677809095977 %i,
+--R 8.1712414612411521E-14 - 6.2831853071794015 %i]
--R ,
--R
--R [1. + 7.5999999999999996 %i, - 10.005039426790001 + 8.5883535709619991
%i,
@@ -2269,8 +2275,8 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R ,
--R
--R [1. + 8.1999999999999993 %i, - 10.909524269378 + 9.8291305671620002 %i,
---R - 10.909524269378375 - 2.7372400471974712 %i,
---R - 3.7481129311345285E-13 - 12.566370614359471 %i]
+--R - 10.909524269378373 - 2.7372400471974712 %i,
+--R - 3.730349362740526E-13 - 12.566370614359471 %i]
--R ,
--R
--R [1. + 8.3000000000000007 %i, - 11.060543221792001 + 10.040273897180001
%i,
@@ -2365,6 +2371,890 @@ log(Gamma(1. + 10.0 * %i))-(- 13.637732188247 +
13.802912974230 * %i)]]
--R Type: List List Complex
DoubleFloat
--E 4
+@
+\section{log(Gamma) 1+0.0i to 1+10.0i by 0+0.1i}
+As you can see from the above, Axiom's value diverges from the published
+values in Abramowitz at 1+4.6i and above. Mathematica shows the same
+divergence at the same place.
+
+\subsection{Abramowitz 6.1.40 equation}
+We can compute the values in Abramowitz from the formula 6.1.40 on
+page 257 which reads:
+\begin{equation}
+ln \Gamma{(z)}\approx(z-\frac{1}{2})ln z-z+\frac{1}{2}ln{(2\pi)}
++\sum_{m=1}^\infty {\frac{B_{2m}}{2m(2m-1)z^{2m-1}}} \quad
+(z\rightarrow \infty in \vert arg z\vert < \pi)
+\end{equation}
+First we compute the constant
+<<*>>=
+--S 5 of 10
+halfLog2Pi:=log(2.0*%pi)/2
+--R
+--R
+--R (5) 0.9189385332 0467274178
+--R Type:
Float
+--E 5
+
+@
+Next we compute the Bernoulli numbers given by the double-sum formula
+\begin{equation}
+B_n=\sum_{k=0}^n{\frac{1}{(k+1)}}\sum_{r=0}^k{(-1)^r\binom{k}{r}r^n}
+\end{equation}
+The inner sum is given by
+<<*>>=
+--S 6 of 10
+inner(k,n)==reduce(+,[(-1)^r*binomial(k,r)*r^n for r in 0..k])
+--R
+--R Type:
Void
+--E 6
+
+@
+and the bernoulli numbers are given by
+<<*>>=
+--S 7 of 10
+B(n)==reduce(+,[(inner(k,n)/(k+1)) for k in 0..n])
+--R
+--R Type:
Void
+--E 7
+
+@
+Now we need to compute the values of a single term in the expansion
+<<*>>=
+--S 8 of 10
+Z(m,z)==B(2*m)/((2*m*(2*m-1))*z^(2*m-1))
+--R
+--R Type:
Void
+--E 8
+
+@
+and we can compute the formula 6.1.41
+<<*>>=
+--S 9 of 10
+H(z)==(z-1/2)*log(z)-z+halfLog2Pi+reduce(+,[Z(m,z) for m in 1..5])
+--R
+--R Type:
Void
+--E 9
+
+@
+The most accurate values appear to be given when 5 terms are used.
+Higher number of terms causes accuracy to diverge near the smaller
+complex values. As you can see this formula reproduces the table
+values in Abramowitz smoothly.
+<<*>>=
+--S 10 of 10
+[[1. + 0.0 * %i,0.,H(1. + 0.0 * %i),H(1. + 0.0 * %i)-0.0],_
+[1. + 0.1 * %i, -0.008197780565 - 0.057322940417 * %i,_
+H(1. + 0.1 * %i),_
+H(1. + 0.1 * %i)-( -0.008197780565 - 0.057322940417 * %i)],_
+[1. + 0.2 * %i, -0.032476292318 - 0.112302222644 * %i,_
+H(1. + 0.2 * %i),_
+H(1. + 0.2 * %i)-( -0.032476292318 - 0.112302222644 * %i)],_
+[1. + 0.3 * %i, -0.071946250900 - 0.162820672168 * %i,_
+H(1. + 0.3 * %i),_
+H(1. + 0.3 * %i)-( -0.071946250900 - 0.162820672168 * %i)],_
+[1. + 0.4 * %i, -0.125289374821 - 0.207155826316 * %i,_
+H(1. + 0.4 * %i),_
+H(1. + 0.4 * %i)-( -0.125289374821 - 0.207155826316 * %i)],_
+[1. + 0.5 * %i,- 0.190945499187 - 0.244058298905 * %i,_
+H(1. + 0.5 * %i),_
+H(1. + 0.5 * %i)-(- 0.190945499187 - 0.244058298905 * %i)],_
+[1. + 0.6 * %i,- 0.267290068214 - 0.272743810491 * %i,_
+H(1. + 0.6 * %i),_
+H(1. + 0.6 * %i)-(- 0.267290068214 - 0.272743810491 * %i)],_
+[1. + 0.7 * %i,- 0.352768690860 - 0.292826351187 * %i,_
+H(1. + 0.7 * %i),_
+H(1. + 0.7 * %i)-(- 0.352768690860 - 0.292826351187 * %i)],_
+[1. + 0.8 * %i,- 0.445978783549 - 0.304225602976 * %i,_
+H(1. + 0.8 * %i),_
+H(1. + 0.8 * %i)-(- 0.445978783549 - 0.304225602976 * %i)],_
+[1. + 0.9 * %i,- 0.545705128605 - 0.307074375642 * %i,_
+H(1. + 0.9 * %i),_
+H(1. + 0.9 * %i)-(- 0.545705128605 - 0.307074375642 * %i)],_
+[1. + 1.0 * %i,- 0.650923199302 - 0.301640320468 * %i,_
+H(1. + 1.0 * %i),_
+H(1. + 1.0 * %i)-(- 0.650923199302 - 0.301640320468 * %i)],_
+[1. + 1.1 * %i,- 0.760783958841 - 0.288266614239 * %i,_
+H(1. + 1.1 * %i),_
+H(1. + 1.1 * %i)-(- 0.760783958841 - 0.288266614239 * %i)],_
+[1. + 1.2 * %i,- 0.874590463895 - 0.267330580581 * %i,_
+H(1. + 1.2 * %i),_
+H(1. + 1.2 * %i)-(- 0.874590463895 - 0.267330580581 * %i)],_
+[1. + 1.3 * %i,- 0.991772766959 - 0.239216784465 * %i,_
+H(1. + 1.3 * %i),_
+H(1. + 1.3 * %i)-(- 0.991772766959 - 0.239216784465 * %i)],_
+[1. + 1.4 * %i,- 1.111864566426 - 0.204300724149 * %i,_
+H(1. + 1.4 * %i),_
+H(1. + 1.4 * %i)-(- 1.111864566426 - 0.204300724149 * %i)],_
+[1. + 1.5 * %i,- 1.234483051547 - 0.162939769480 * %i,_
+H(1. + 1.5 * %i),_
+H(1. + 1.5 * %i)-(- 1.234483051547 - 0.162939769480 * %i)],_
+[1. + 1.6 * %i,- 1.359312248465 - 0.115468793589 * %i,_
+H(1. + 1.6 * %i),_
+H(1. + 1.6 * %i)-(- 1.359312248465 - 0.115468793589 * %i)],_
+[1. + 1.7 * %i,- 1.486089612757 - 0.062198698329 * %i,_
+H(1. + 1.7 * %i),_
+H(1. + 1.7 * %i)-(- 1.486089612757 - 0.062198698329 * %i)],_
+[1. + 1.8 * %i,- 1.614595396000 - 0.003416631477 * %i,_
+H(1. + 1.8 * %i),_
+H(1. + 1.8 * %i)-(- 1.614595396000 - 0.003416631477 * %i)],_
+[1. + 1.9 * %i,- 1.744644276174 + 0.060612874295 * %i,_
+H(1. + 1.9 * %i),_
+H(1. + 1.9 * %i)-(- 1.744644276174 + 0.060612874295 * %i)],_
+[1. + 2.0 * %i,- 1.876078786431 + 0.129646316310 * %i,_
+H(1. + 2.0 * %i),_
+H(1. + 2.0 * %i)-(- 1.876078786431 + 0.129646316310 * %i)],_
+[1. + 2.1 * %i,- 2.008764150471 + 0.203459473833 * %i,_
+H(1. + 2.1 * %i),_
+H(1. + 2.1 * %i)-(- 2.008764150471 + 0.203459473833 * %i)],_
+[1. + 2.2 * %i,- 2.142584209296 + 0.281845658426 * %i,_
+H(1. + 2.2 * %i),_
+H(1. + 2.2 * %i)-(- 2.142584209296 + 0.281845658426 * %i)],_
+[1. + 2.3 * %i,- 2.277438192204 + 0.364614048950 * %i,_
+H(1. + 2.3 * %i),_
+H(1. + 2.3 * %i)-(- 2.277438192204 + 0.364614048950 * %i)],_
+[1. + 2.4 * %i,- 2.413238141184 + 0.451588152441 * %i,_
+H(1. + 2.4 * %i),_
+H(1. + 2.4 * %i)-(- 2.413238141184 + 0.451588152441 * %i)],_
+[1. + 2.5 * %i,- 2.549906842495 + 0.542604405852 * %i,_
+H(1. + 2.5 * %i),_
+H(1. + 2.5 * %i)-(- 2.549906842495 + 0.542604405852 * %i)],_
+[1. + 2.6 * %i,- 2.687376153750 + 0.637510919046 * %i,_
+H(1. + 2.6 * %i),_
+H(1. + 2.6 * %i)-(- 2.687376153750 + 0.637510919046 * %i)],_
+[1. + 2.7 * %i,- 2.825585641191 + 0.736166351679 * %i,_
+H(1. + 2.7 * %i),_
+H(1. + 2.7 * %i)-(- 2.825585641191 + 0.736166351679 * %i)],_
+[1. + 2.8 * %i,- 2.964481461789 + 0.838438913096 * %i,_
+H(1. + 2.8 * %i),_
+H(1. + 2.8 * %i)-(- 2.964481461789 + 0.838438913096 * %i)],_
+[1. + 2.9 * %i,- 3.104015439901 + 0.944205473039 * %i,_
+H(1. + 2.9 * %i),_
+H(1. + 2.9 * %i)-(- 3.104015439901 + 0.944205473039 * %i)],_
+[1. + 3.0 * %i,- 3.244144299590 + 1.053350771069 * %i,_
+H(1. + 3.0 * %i),_
+H(1. + 3.0 * %i)-(- 3.244144299590 + 1.053350771069 * %i)],_
+[1. + 3.1 * %i,- 3.384829022377 + 1.165766713286 * %i,_
+H(1. + 3.1 * %i),_
+H(1. + 3.1 * %i)-(- 3.384829022377 + 1.165766713286 * %i)],_
+[1. + 3.2 * %i,- 3.526034306709 + 1.281351745932 * %i,_
+H(1. + 3.2 * %i),_
+H(1. + 3.2 * %i)-(- 3.526034306709 + 1.281351745932 * %i)],_
+[1. + 3.3 * %i,- 3.667728110488 + 1.400010296576 * %i,_
+H(1. + 3.3 * %i),_
+H(1. + 3.3 * %i)-(- 3.667728110488 + 1.400010296576 * %i)],_
+[1. + 3.4 * %i,- 3.809881261823 + 1.521652274673 * %i,_
+H(1. + 3.4 * %i),_
+H(1. + 3.4 * %i)-(- 3.809881261823 + 1.521652274673 * %i)],_
+[1. + 3.5 * %i,- 3.952467126189 + 1.646192624269 * %i,_
+H(1. + 3.5 * %i),_
+H(1. + 3.5 * %i)-(- 3.952467126189 + 1.646192624269 * %i)],_
+[1. + 3.6 * %i,- 4.095461320451 + 1.773550922591 * %i,_
+H(1. + 3.6 * %i),_
+H(1. + 3.6 * %i)-(- 4.095461320451 + 1.773550922591 * %i)],_
+[1. + 3.7 * %i,- 4.238841466071 + 1.903651019019 * %i,_
+H(1. + 3.7 * %i),_
+H(1. + 3.7 * %i)-(- 4.238841466071 + 1.903651019019 * %i)],_
+[1. + 3.8 * %i,- 4.382586975228 + 2.036420709693 * %i,_
+H(1. + 3.8 * %i),_
+H(1. + 3.8 * %i)-(- 4.382586975228 + 2.036420709693 * %i)],_
+[1. + 3.9 * %i,- 4.526678864716 + 2.171791443605 * %i,_
+H(1. + 3.9 * %i),_
+H(1. + 3.9 * %i)-(- 4.526678864716 + 2.171791443605 * %i)],_
+[1. + 4.0 * %i,- 4.671099593409 + 2.309698056573 * %i,_
+H(1. + 4.0 * %i),_
+H(1. + 4.0 * %i)-(- 4.671099593409 + 2.309698056573 * %i)],_
+[1. + 4.1 * %i,- 4.815832919796 + 2.450078529947 * %i,_
+H(1. + 4.1 * %i),_
+H(1. + 4.1 * %i)-(- 4.815832919796 + 2.450078529947 * %i)],_
+[1. + 4.2 * %i,- 4.960863776687 + 2.592873771319 * %i,_
+H(1. + 4.2 * %i),_
+H(1. + 4.2 * %i)-(- 4.960863776687 + 2.592873771319 * %i)],_
+[1. + 4.3 * %i,- 5.106178160663 + 2.738027414820 * %i,_
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+H(1. + 4.3 * %i)-(- 5.106178160663 + 2.738027414820 * %i)],_
+[1. + 4.4 * %i,- 5.251763034230 + 2.885485638927 * %i,_
+H(1. + 4.4 * %i),_
+H(1. + 4.4 * %i)-(- 5.251763034230 + 2.885485638927 * %i)],_
+[1. + 4.5 * %i,- 5.397606238984 + 3.035196999922 * %i,_
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+[1. + 4.6 * %i,- 5.543696418304 + 3.187112279389 * %i,_
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+[1. + 4.7 * %i,- 5.690022948373 + 3.341184344327 * %i,_
+H(1. + 4.7 * %i),_
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+[1. + 4.8 * %i,- 5.836575876454 + 3.497368018615 * %i,_
+H(1. + 4.8 * %i),_
+H(1. + 4.8 * %i)-(- 5.836575876454 + 3.497368018615 * %i)],_
+[1. + 4.9 * %i,- 5.983345865532 + 3.655619964712 * %i,_
+H(1. + 4.9 * %i),_
+H(1. + 4.9 * %i)-(- 5.983345865532 + 3.655619964712 * %i)],_
+[1. + 5.0 * %i,- 6.130324144553 + 3.815898574615 * %i,_
+H(1. + 5.0 * %i),_
+H(1. + 5.0 * %i)-(- 6.130324144553 + 3.815898574615 * %i)],_
+[1. + 5.1 * %i,- 6.277502463584 + 3.978163869188 * %i,_
+H(1. + 5.1 * %i),_
+H(1. + 5.1 * %i)-(- 6.277502463584 + 3.978163869188 * %i)],_
+[1. + 5.2 * %i,- 6.424873053335 + 4.142377405086 * %i,_
+H(1. + 5.2 * %i),_
+H(1. + 5.2 * %i)-(- 6.424873053335 + 4.142377405086 * %i)],_
+[1. + 5.3 * %i,- 6.572428588529 + 4.308502188583 * %i,_
+H(1. + 5.3 * %i),_
+H(1. + 5.3 * %i)-(- 6.572428588529 + 4.308502188583 * %i)],_
+[1. + 5.4 * %i,- 6.720162154703 + 4.476502595668 * %i,_
+H(1. + 5.4 * %i),_
+H(1. + 5.4 * %i)-(- 6.720162154703 + 4.476502595668 * %i)],_
+[1. + 5.5 * %i,- 6.868067218048 + 4.646344297870 * %i,_
+H(1. + 5.5 * %i),_
+H(1. + 5.5 * %i)-(- 6.868067218048 + 4.646344297870 * %i)],_
+[1. + 5.6 * %i,- 7.016137597976 + 4.817994193305 * %i,_
+H(1. + 5.6 * %i),_
+H(1. + 5.6 * %i)-(- 7.016137597976 + 4.817994193305 * %i)],_
+[1. + 5.7 * %i,- 7.164367442106 + 4.991420342489 * %i,_
+H(1. + 5.7 * %i),_
+H(1. + 5.7 * %i)-(- 7.164367442106 + 4.991420342489 * %i)],_
+[1. + 5.8 * %i,- 7.312751203430 + 5.166591908537 * %i,_
+H(1. + 5.8 * %i),_
+H(1. + 5.8 * %i)-(- 7.312751203430 + 5.166591908537 * %i)],_
+[1. + 5.9 * %i,- 7.461283619429 + 5.343479101353 * %i,_
+H(1. + 5.9 * %i),_
+H(1. + 5.9 * %i)-(- 7.461283619429 + 5.343479101353 * %i)],_
+[1. + 6.0 * %i,- 7.609959692951 + 5.522053125515 * %i,_
+H(1. + 6.0 * %i),_
+H(1. + 6.0 * %i)-(- 7.609959692951 + 5.522053125515 * %i)],_
+[1. + 6.1 * %i,- 7.758774674655 + 5.702286131535 * %i,_
+H(1. + 6.1 * %i),_
+H(1. + 6.1 * %i)-(- 7.758774674655 + 5.702286131535 * %i)],_
+[1. + 6.2 * %i,- 7.907724046898 + 5.884151170239 * %i,_
+H(1. + 6.2 * %i),_
+H(1. + 6.2 * %i)-(- 7.907724046898 + 5.884151170239 * %i)],_
+[1. + 6.3 * %i,- 8.056803508904 + 6.067622150013 * %i,_
+H(1. + 6.3 * %i),_
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+[1. + 6.4 * %i,- 8.206008963100 + 6.252673796705 * %i,_
+H(1. + 6.4 * %i),_
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+[1. + 6.5 * %i,- 8.355336502511 + 6.439281615976 * %i,_
+H(1. + 6.5 * %i),_
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+[1. + 6.6 * %i,- 8.504782399125 + 6.627421857912 * %i,_
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+[1. + 6.7 * %i,- 8.654343093123 + 6.817071483744 * %i,_
+H(1. + 6.7 * %i),_
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+[1. + 6.8 * %i,- 8.804015182910 + 7.008208134502 * %i,_
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+[1. + 6.9 * %i,- 8.953795415879 + 7.200810101493 * %i,_
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+[1. + 7.0 * %i,- 9.103680679832 + 7.394856298436 * %i,_
+H(1. + 7.0 * %i),_
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+[1. + 7.1 * %i,- 9.253667995015 + 7.590326235184 * %i,_
+H(1. + 7.1 * %i),_
+H(1. + 7.1 * %i)-(- 9.253667995015 + 7.590326235184 * %i)],_
+[1. + 7.2 * %i,- 9.403754506708 + 7.787199992877 * %i,_
+H(1. + 7.2 * %i),_
+H(1. + 7.2 * %i)-(- 9.403754506708 + 7.787199992877 * %i)],_
+[1. + 7.3 * %i,- 9.553937478321 + 7.985458200468 * %i,_
+H(1. + 7.3 * %i),_
+H(1. + 7.3 * %i)-(- 9.553937478321 + 7.985458200468 * %i)],_
+[1. + 7.4 * %i,- 9.704214284972 + 8.185082012503 * %i,_
+H(1. + 7.4 * %i),_
+H(1. + 7.4 * %i)-(- 9.704214284972 + 8.185082012503 * %i)],_
+[1. + 7.5 * %i,- 9.854582407486 + 8.386053088089 * %i,_
+H(1. + 7.5 * %i),_
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+[1. + 7.6 * %i,- 10.005039426790 + 8.588353570962 * %i,_
+H(1. + 7.6 * %i),_
+H(1. + 7.6 * %i)-(- 10.005039426790 + 8.588353570962 * %i)],_
+[1. + 7.7 * %i,- 10.155583018686 + 8.791966070587 * %i,_
+H(1. + 7.7 * %i),_
+H(1. + 7.7 * %i)-(- 10.155583018686 + 8.791966070587 * %i)],_
+[1. + 7.8 * %i,- 10.306210948948 + 8.996873644229 * %i,_
+H(1. + 7.8 * %i),_
+H(1. + 7.8 * %i)-(- 10.306210948948 + 8.996873644229 * %i)],_
+[1. + 7.9 * %i,- 10.456921068739 + 9.203059779925 * %i,_
+H(1. + 7.9 * %i),_
+H(1. + 7.9 * %i)-(- 10.456921068739 + 9.203059779925 * %i)],_
+[1. + 8.0 * %i,- 10.607711310315 + 9.410508380312 * %i,_
+H(1. + 8.0 * %i),_
+H(1. + 8.0 * %i)-(- 10.607711310315 + 9.410508380312 * %i)],_
+[1. + 8.1 * %i,- 10.758579682995 + 9.619203747242 * %i,_
+H(1. + 8.1 * %i),_
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+[1. + 8.2 * %i,- 10.909524269378 + 9.829130567162 * %i,_
+H(1. + 8.2 * %i),_
+H(1. + 8.2 * %i)-(- 10.909524269378 + 9.829130567162 * %i)],_
+[1. + 8.3 * %i,- 11.060543221792 + 10.040273897180 * %i,_
+H(1. + 8.3 * %i),_
+H(1. + 8.3 * %i)-(- 11.060543221792 + 10.040273897180 * %i)],_
+[1. + 8.4 * %i,- 11.211634758948 + 10.252619151809 * %i,_
+H(1. + 8.4 * %i),_
+H(1. + 8.4 * %i)-(- 11.211634758948 + 10.252619151809 * %i)],_
+[1. + 8.5 * %i,- 11.362797162804 + 10.466152090324 * %i,_
+H(1. + 8.5 * %i),_
+H(1. + 8.5 * %i)-(- 11.362797162804 + 10.466152090324 * %i)],_
+[1. + 8.6 * %i,- 11.514028775602 + 10.680858804712 * %i,_
+H(1. + 8.6 * %i),_
+H(1. + 8.6 * %i)-(- 11.514028775602 + 10.680858804712 * %i)],_
+[1. + 8.7 * %i,- 11.665327997081 + 10.896725708177 * %i,_
+H(1. + 8.7 * %i),_
+H(1. + 8.7 * %i)-(- 11.665327997081 + 10.896725708177 * %i)],_
+[1. + 8.8 * %i,- 11.816693281848 + 11.113739524157 * %i,_
+H(1. + 8.8 * %i),_
+H(1. + 8.8 * %i)-(- 11.816693281848 + 11.113739524157 * %i)],_
+[1. + 8.9 * %i,- 11.968123136901 + 11.331887275853 * %i,_
+H(1. + 8.9 * %i),_
+H(1. + 8.9 * %i)-(- 11.968123136901 + 11.331887275853 * %i)],_
+[1. + 9.0 * %i,- 12.119616119281 + 11.551156276202 * %i,_
+H(1. + 9.0 * %i),_
+H(1. + 9.0 * %i)-(- 12.119616119281 + 11.551156276202 * %i)],_
+[1. + 9.1 * %i,- 12.271170833867 + 11.771534118309 * %i,_
+H(1. + 9.1 * %i),_
+H(1. + 9.1 * %i)-(- 12.271170833867 + 11.771534118309 * %i)],_
+[1. + 9.2 * %i,- 12.422785931281 + 11.993008666285 * %i,_
+H(1. + 9.2 * %i),_
+H(1. + 9.2 * %i)-(- 12.422785931281 + 11.993008666285 * %i)],_
+[1. + 9.3 * %i,- 12.574460105908 + 12.215568046479 * %i,_
+H(1. + 9.3 * %i),_
+H(1. + 9.3 * %i)-(- 12.574460105908 + 12.215568046479 * %i)],_
+[1. + 9.4 * %i,- 12.726192094029 + 12.439200639090 * %i,_
+H(1. + 9.4 * %i),_
+H(1. + 9.4 * %i)-(- 12.726192094029 + 12.439200639090 * %i)],_
+[1. + 9.5 * %i,- 12.877980672044 + 12.663895070128 * %i,_
+H(1. + 9.5 * %i),_
+H(1. + 9.5 * %i)-(- 12.877980672044 + 12.663895070128 * %i)],_
+[1. + 9.6 * %i,- 13.029824654789 + 12.889640203708 * %i,_
+H(1. + 9.6 * %i),_
+H(1. + 9.6 * %i)-(- 13.029824654789 + 12.889640203708 * %i)],_
+[1. + 9.7 * %i,- 13.181722893951 + 13.116425134666 * %i,_
+H(1. + 9.7 * %i),_
+H(1. + 9.7 * %i)-(- 13.181722893951 + 13.116425134666 * %i)],_
+[1. + 9.8 * %i,- 13.333674276547 + 13.344239181477 * %i,_
+H(1. + 9.8 * %i),_
+H(1. + 9.8 * %i)-(- 13.333674276547 + 13.344239181477 * %i)],_
+[1. + 9.9 * %i,- 13.485677723495 + 13.573071879455 * %i,_
+H(1. + 9.9 * %i),_
+H(1. + 9.9 * %i)-(- 13.485677723495 + 13.573071879455 * %i)],_
+[1. + 10.0 * %i,- 13.637732188247 + 13.802912974230 * %i,_
+H(1. + 10.0 * %i),_
+H(1. + 10.0 * %i)-(- 13.637732188247 + 13.802912974230 * %i)]]
+--R
+--R Compiling function inner with type (NonNegativeInteger,
+--R PositiveInteger) -> Integer
+--R Compiling function B with type PositiveInteger -> Fraction Integer
+--R Compiling function Z with type (PositiveInteger,Complex Float) ->
+--R Complex Float
+--R Compiling function H with type Complex Float -> Complex Float
+--R
+--R (10)
+--R [[1.0,0.0,0.0005342523 0039183749 9,0.0005342523 0039183749 9],
+--R
+--R [1.0 + 0.1 %i, - 0.0081977805 65 - 0.0573229404 17 %i,
+--R - 0.0079060360 5641775235 1 - 0.0577425561 880608609 %i,
+--R 0.0002917445 0858224764 9 - 0.0004196157 710608609 %i]
+--R ,
+--R
+--R [1.0 + 0.2 %i, - 0.0324762923 18 - 0.1123022226 44 %i,
+--R - 0.0326253424 137575225 - 0.1127252606 10617179 %i,
+--R - 0.0001490500 957575225 - 0.0004230379 66617179 %i]
+--R ,
+--R
+--R [1.0 + 0.3 %i, - 0.0719462509 - 0.1628206721 68 %i,
+--R - 0.0722917796 3325300528 - 0.1629343118 0182595104 %i,
+--R - 0.0003455287 3325300528 - 0.0001136396 33825951 %i]
+--R ,
+--R
+--R [1.0 + 0.4 %i, - 0.1252893748 21 - 0.2071558263 16 %i,
+--R - 0.1255237832 8903895364 - 0.2070111768 1461131296 %i,
+--R - 0.0002344084 6803895364 + 0.0001446495 0138868704 %i]
+--R ,
+--R
+--R [1.0 + 0.5 %i, - 0.1909454991 87 - 0.2440582989 05 %i,
+--R - 0.1909844153 9406580427 - 0.2438650024 4353127544 %i,
+--R - 0.0000389162 0706580427 + 0.0001932964 6146872456 %i]
+--R ,
+--R
+--R [1.0 + 0.6 %i, - 0.2672900682 14 - 0.2727438104 91 %i,
+--R - 0.2672177072 7708327583 - 0.2726297345 2049521095 %i,
+--R 0.0000723609 3691672417 + 0.0001140759 70504789 %i]
+--R ,
+--R
+--R [1.0 + 0.7 %i, - 0.3527686908 6 - 0.2928263511 87 %i,
+--R - 0.3526830217 4071123327 - 0.2928000766 8168133466 %i,
+--R 0.0000856691 1928876673 + 0.0000262745 053186653 %i]
+--R ,
+--R
+--R [1.0 + 0.8 %i, - 0.4459787835 49 - 0.3042256029 76 %i,
+--R - 0.4459243064 600936154 - 0.3042458540 6494479039 %i,
+--R 0.0000544770 889063846 - 0.0000202510 889447904 %i]
+--R ,
+--R
+--R [1.0 + 0.9 %i, - 0.5457051286 05 - 0.3070743756 42 %i,
+--R - 0.5456837564 2266021862 - 0.3071047743 7404541115 %i,
+--R 0.0000213721 823397814 - 0.0000303987 320454111 %i]
+--R ,
+--R
+--R [1.0 + %i, - 0.6509231993 02 - 0.3016403204 68 %i,
+--R - 0.6509218279 5803214573 - 0.3016638509 8907615826 %i,
+--R 0.0000013713 439678543 - 0.0000235305 210761583 %i]
+--R ,
+--R
+--R [1.0 + 1.1 %i, - 0.7607839588 41 - 0.2882666142 39 %i,
+--R - 0.7607904806 1959258337 - 0.2882800153 6783960204 %i,
+--R - 0.0000065217 7859258337 - 0.0000134011 28839602 %i]
+--R ,
+--R
+--R [1.0 + 1.2 %i, - 0.8745904638 95 - 0.2673305805 81 %i,
+--R - 0.8745979912 3533931918 - 0.2673362568 721629649 %i,
+--R - 0.0000075273 4033931918 - 0.0000056762 911629649 %i]
+--R ,
+--R
+--R [1.0 + 1.3 %i, - 0.9917727669 59 - 0.2392167844 65 %i,
+--R - 0.9917786169 8448669121 - 0.2392180306 8558905783 %i,
+--R - 0.0000058500 2548669121 - 0.0000012462 2058905783 %i]
+--R ,
+--R
+--R [1.0 + 1.4 %i, - 1.1118645664 26 - 0.2043007241 49 %i,
+--R - 1.1118683074 739622893 - 0.2042999880 7236175351 %i,
+--R - 0.0000037410 479622893 + 0.7360766382 4649 E -6 %i]
+--R ,
+--R
+--R [1.0 + 1.5 %i, - 1.2344830515 47 - 0.1629397694 8 %i,
+--R - 1.2344851106 088582896 - 0.1629384511 4283237552 %i,
+--R - 0.0000020590 618582896 + 0.0000013183 3716762448 %i]
+--R ,
+--R
+--R [1.0 + 1.6 %i, - 1.3593122484 65 - 0.1154687935 89 %i,
+--R - 1.3593132065 575271954 - 0.1154675392 0060867755 %i,
+--R - 0.9580925271 9539 E -6 + 0.0000012543 8839132245 %i]
+--R ,
+--R
+--R [1.0 + 1.7 %i, - 1.4860896127 57 - 0.0621986983 29 %i,
+--R - 1.4860899424 768375682 - 0.0621977263 0645742299 5 %i,
+--R - 0.3297198375 682 E -6 + 0.9720225425 77005 E -6 %i]
+--R ,
+--R
+--R [1.0 + 1.8 %i, - 1.614595396 - 0.0034166314 77 %i,
+--R - 1.6145954117 853473727 - 0.0034159591 4562980466 %i,
+--R - 0.1578534737 27 E -7 + 0.6723313701 953421 E -6 %i]
+--R ,
+--R
+--R [1.0 + 1.9 %i, - 1.7446442761 74 + 0.0606128742 95 %i,
+--R - 1.7446441619 196944916 + 0.0606133033 8564864100 7 %i,
+--R 0.1142543055 08 E -6 + 0.4290906486 41007 E -6 %i]
+--R ,
+--R
+--R [1.0 + 2.0 %i, - 1.8760787864 31 + 0.1296463163 1 %i,
+--R - 1.8760786381 585810542 + 0.1296465718 833341783 %i,
+--R 0.1482724189 458 E -6 + 0.2555733341 783 E -6 %i]
+--R ,
+--R
+--R [1.0 + 2.1 %i, - 2.0087641504 71 + 0.2034594738 33 %i,
+--R - 2.0087640119 218530981 + 0.2034596154 799745979 %i,
+--R 0.1385491469 02 E -6 + 0.1416469745 979 E -6 %i]
+--R ,
+--R
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+--E 10
)spool
)lisp (bye)
- [Axiom-developer] 20080221.01.tpd.patch (7099: complex gamma function investigation),
daly <=