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[Axiom-developer] 20080222.01.tpd.patch (7099 logGamma vs log(Gamma) )
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From: |
daly |
|
Subject: |
[Axiom-developer] 20080222.01.tpd.patch (7099 logGamma vs log(Gamma) ) |
|
Date: |
Thu, 21 Feb 2008 21:47:50 -0600 |
More investigation of the Gamma, log(Gamma) and logGamma functions.
======================================================================
diff --git a/changelog b/changelog
index eea576d..c7ce080 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,4 @@
+20080222 tpd src/input/gamma.input 7099 logGamma vs log(Gamma)
20080221 wxh src/interp/sfsfun.boot increase precision of PI (236)
20080221 tpd src/input/gamma.input investigate complex gamma issues
20080220 tpd src/hyper/bookvol11 add additional hyperdoc page translations
diff --git a/src/input/gamma.input.pamphlet b/src/input/gamma.input.pamphlet
index a8c533c..6ccd4a5 100644
--- a/src/input/gamma.input.pamphlet
+++ b/src/input/gamma.input.pamphlet
@@ -26,7 +26,7 @@ Dover Publications, Inc. New York 1965. pp267-270
@
\section{Gamma 1.000 to 1.995 by 0.005}
<<*>>=
---S 1 of 10
+--S 1 of 12
[[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],_
@@ -759,8 +759,10 @@ Dover Publications, Inc. New York 1965. pp267-270
--R Type: List List
DoubleFloat
--E 1
+@
\section{Psi}
---S 2 of 10
+<<*>>=
+--S 2 of 12
Psi(x:DFLOAT):DFLOAT==polygamma(0,x)
--R
--R Function declaration Psi : DoubleFloat -> DoubleFloat has been added
@@ -777,7 +779,7 @@ Dover Publications, Inc. New York 1965. pp267-270
\section{Psi 1.000 to 2.000 by 0.005}
<<*>>=
---S 3 of 10
+--S 3 of 12
[[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],_
@@ -1551,7 +1553,7 @@ 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 10
+--S 4 of 12
[[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)),_
@@ -2387,7 +2389,7 @@ ln \Gamma{(z)}\approx(z-\frac{1}{2})ln
z-z+\frac{1}{2}ln{(2\pi)}
\end{equation}
First we compute the constant
<<*>>=
---S 5 of 10
+--S 5 of 12
halfLog2Pi:=log(2.0*%pi)/2
--R
--R
@@ -2402,7 +2404,7 @@
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
+--S 6 of 12
inner(k,n)==reduce(+,[(-1)^r*binomial(k,r)*r^n for r in 0..k])
--R
--R Type:
Void
@@ -2411,7 +2413,7 @@ inner(k,n)==reduce(+,[(-1)^r*binomial(k,r)*r^n for r in
0..k])
@
and the bernoulli numbers are given by
<<*>>=
---S 7 of 10
+--S 7 of 12
B(n)==reduce(+,[(inner(k,n)/(k+1)) for k in 0..n])
--R
--R Type:
Void
@@ -2420,7 +2422,7 @@ B(n)==reduce(+,[(inner(k,n)/(k+1)) for k in 0..n])
@
Now we need to compute the values of a single term in the expansion
<<*>>=
---S 8 of 10
+--S 8 of 12
Z(m,z)==B(2*m)/((2*m*(2*m-1))*z^(2*m-1))
--R
--R Type:
Void
@@ -2429,7 +2431,7 @@ Z(m,z)==B(2*m)/((2*m*(2*m-1))*z^(2*m-1))
@
and we can compute the formula 6.1.41
<<*>>=
---S 9 of 10
+--S 9 of 12
H(z)==(z-1/2)*log(z)-z+halfLog2Pi+reduce(+,[Z(m,z) for m in 1..5])
--R
--R Type:
Void
@@ -2441,7 +2443,7 @@ 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
+--S 10 of 12
[[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),_
@@ -3255,6 +3257,1661 @@ H(1. + 10.0 * %i)-(- 13.637732188247 + 13.802912974230
* %i)]]
--R ]
--R Type: List List Complex
Float
--E 10
+@
+Another useful definition of the logGamma function that is
+continuous across the cut [1+4.5i,1+4.6i] is:
+<<*>>=
+--S 11 of 12
+lng2(xx:COMPLEX(DFLOAT)):COMPLEX(DFLOAT)==
+ y:COMPLEX(DFLOAT):=xx;
+ x:COMPLEX(DFLOAT):=xx;
+ t1:COMPLEX(DFLOAT):=x+5.5-(x+0.5)*log(x+5.5)
+ ser:COMPLEX(DFLOAT):=1.000000000190015
+ y:=y+1;
+ ser:=ser+(76.18009172947146/y)
+ y:=y+1;
+ ser:=ser+(-86.50532032941677/y)
+ y:=y+1;
+ ser:=ser+(24.01409824083091/y)
+ y:=y+1;
+ ser:=ser+(-1.231739572450155/y)
+ y:=y+1;
+ ser:=ser+(0.1208650973866179E-2/y)
+ y:=y+1;
+ ser:=ser+(-0.5395239384953E-5/y)
+ result:COMPLEX(DFLOAT):=log(2.5066282746310005*ser/x)-t1
+ result
+--R
+--R Function declaration lng2 : Complex DoubleFloat -> Complex
+--R DoubleFloat has been added to workspace.
+--R Type:
Void
+--E 11
+
+@
+and we can compare this with the log(Gamma) and logGamma functions in
+Axiom:
+<<*>>=
+--S 12 of 12
+[[1. + 0.0 * %i,0.,lng2(1. + 0.0 * %i),lng2(1. + 0.0 * %i)-0.0],_
+[1. + 0.1 * %i, -0.008197780565 - 0.057322940417 * %i,_
+lng2(1. + 0.1 * %i),log(Gamma(1. + 0.1 * %i)),_
+lng2(1. + 0.1 * %i)-log(Gamma(1. + 0.1 * %i)),_
+lng2(1. + 0.1 * %i),logGamma(1. + 0.1 * %i),_
+lng2(1. + 0.1 * %i)-logGamma(1. + 0.1 * %i),_
+lng2(1. + 0.1 * %i)-( -0.008197780565 - 0.057322940417 * %i)],_
+[1. + 0.2 * %i, -0.032476292318 - 0.112302222644 * %i,_
+lng2(1. + 0.2 * %i),log(Gamma(1. + 0.2 * %i)),_
+lng2(1. + 0.2 * %i)-log(Gamma(1. + 0.2 * %i)),_
+lng2(1. + 0.2 * %i),logGamma(1. + 0.2 * %i),_
+lng2(1. + 0.2 * %i)-logGamma(1. + 0.2 * %i),_
+lng2(1. + 0.2 * %i)-( -0.032476292318 - 0.112302222644 * %i)],_
+[1. + 0.3 * %i, -0.071946250900 - 0.162820672168 * %i,_
+lng2(1. + 0.3 * %i),log(Gamma(1. + 0.3 * %i)),_
+lng2(1. + 0.3 * %i)-log(Gamma(1. + 0.3 * %i)),_
+lng2(1. + 0.3 * %i),logGamma(1. + 0.3 * %i),_
+lng2(1. + 0.3 * %i)-logGamma(1. + 0.3 * %i),_
+lng2(1. + 0.3 * %i)-( -0.071946250900 - 0.162820672168 * %i)],_
+[1. + 0.4 * %i, -0.125289374821 - 0.207155826316 * %i,_
+lng2(1. + 0.4 * %i),log(Gamma(1. + 0.4 * %i)),_
+lng2(1. + 0.4 * %i)-log(Gamma(1. + 0.4 * %i)),_
+lng2(1. + 0.4 * %i),logGamma(1. + 0.4 * %i),_
+lng2(1. + 0.4 * %i)-logGamma(1. + 0.4 * %i),_
+lng2(1. + 0.4 * %i)-( -0.125289374821 - 0.207155826316 * %i)],_
+[1. + 0.5 * %i,- 0.190945499187 - 0.244058298905 * %i,_
+lng2(1. + 0.5 * %i),log(Gamma(1. + 0.5 * %i)),_
+lng2(1. + 0.5 * %i)-log(Gamma(1. + 0.5 * %i)),_
+lng2(1. + 0.5 * %i),logGamma(1. + 0.5 * %i),_
+lng2(1. + 0.5 * %i)-logGamma(1. + 0.5 * %i),_
+lng2(1. + 0.5 * %i)-(- 0.190945499187 - 0.244058298905 * %i)],_
+[1. + 0.6 * %i,- 0.267290068214 - 0.272743810491 * %i,_
+lng2(1. + 0.6 * %i),log(Gamma(1. + 0.6 * %i)),_
+lng2(1. + 0.6 * %i)-log(Gamma(1. + 0.6 * %i)),_
+lng2(1. + 0.6 * %i),logGamma(1. + 0.6 * %i),_
+lng2(1. + 0.6 * %i)-logGamma(1. + 0.6 * %i),_
+lng2(1. + 0.6 * %i)-(- 0.267290068214 - 0.272743810491 * %i)],_
+[1. + 0.7 * %i,- 0.352768690860 - 0.292826351187 * %i,_
+lng2(1. + 0.7 * %i),log(Gamma(1. + 0.7 * %i)),_
+lng2(1. + 0.7 * %i)-log(Gamma(1. + 0.7 * %i)),_
+lng2(1. + 0.7 * %i),logGamma(1. + 0.7 * %i),_
+lng2(1. + 0.7 * %i)-logGamma(1. + 0.7 * %i),_
+lng2(1. + 0.7 * %i)-(- 0.352768690860 - 0.292826351187 * %i)],_
+[1. + 0.8 * %i,- 0.445978783549 - 0.304225602976 * %i,_
+lng2(1. + 0.8 * %i),log(Gamma(1. + 0.8 * %i)),_
+lng2(1. + 0.8 * %i)-log(Gamma(1. + 0.8 * %i)),_
+lng2(1. + 0.8 * %i),logGamma(1. + 0.8 * %i),_
+lng2(1. + 0.8 * %i)-logGamma(1. + 0.8 * %i),_
+lng2(1. + 0.8 * %i)-(- 0.445978783549 - 0.304225602976 * %i)],_
+[1. + 0.9 * %i,- 0.545705128605 - 0.307074375642 * %i,_
+lng2(1. + 0.9 * %i),log(Gamma(1. + 0.9 * %i)),_
+lng2(1. + 0.9 * %i)-log(Gamma(1. + 0.9 * %i)),_
+lng2(1. + 0.9 * %i),logGamma(1. + 0.9 * %i),_
+lng2(1. + 0.9 * %i)-logGamma(1. + 0.9 * %i),_
+lng2(1. + 0.9 * %i)-(- 0.545705128605 - 0.307074375642 * %i)],_
+[1. + 1.0 * %i,- 0.650923199302 - 0.301640320468 * %i,_
+lng2(1. + 1.0 * %i),log(Gamma(1. + 1.0 * %i)),_
+lng2(1. + 1.0 * %i)-log(Gamma(1. + 1.0 * %i)),_
+lng2(1. + 1.0 * %i),logGamma(1. + 1.0 * %i),_
+lng2(1. + 1.0 * %i)-logGamma(1. + 1.0 * %i),_
+lng2(1. + 1.0 * %i)-(- 0.650923199302 - 0.301640320468 * %i)],_
+[1. + 1.1 * %i,- 0.760783958841 - 0.288266614239 * %i,_
+lng2(1. + 1.1 * %i),log(Gamma(1. + 1.1 * %i)),_
+lng2(1. + 1.1 * %i)-log(Gamma(1. + 1.1 * %i)),_
+lng2(1. + 1.1 * %i),logGamma(1. + 1.1 * %i),_
+lng2(1. + 1.1 * %i)-logGamma(1. + 1.1 * %i),_
+lng2(1. + 1.1 * %i)-(- 0.760783958841 - 0.288266614239 * %i)],_
+[1. + 1.2 * %i,- 0.874590463895 - 0.267330580581 * %i,_
+lng2(1. + 1.2 * %i),log(Gamma(1. + 1.2 * %i)),_
+lng2(1. + 1.2 * %i)-log(Gamma(1. + 1.2 * %i)),_
+lng2(1. + 1.2 * %i),logGamma(1. + 1.2 * %i),_
+lng2(1. + 1.2 * %i)-logGamma(1. + 1.2 * %i),_
+lng2(1. + 1.2 * %i)-(- 0.874590463895 - 0.267330580581 * %i)],_
+[1. + 1.3 * %i,- 0.991772766959 - 0.239216784465 * %i,_
+lng2(1. + 1.3 * %i),log(Gamma(1. + 1.3 * %i)),_
+lng2(1. + 1.3 * %i)-log(Gamma(1. + 1.3 * %i)),_
+lng2(1. + 1.3 * %i),logGamma(1. + 1.3 * %i),_
+lng2(1. + 1.3 * %i)-logGamma(1. + 1.3 * %i),_
+lng2(1. + 1.3 * %i)-(- 0.991772766959 - 0.239216784465 * %i)],_
+[1. + 1.4 * %i,- 1.111864566426 - 0.204300724149 * %i,_
+lng2(1. + 1.4 * %i),log(Gamma(1. + 1.4 * %i)),_
+lng2(1. + 1.4 * %i)-log(Gamma(1. + 1.4 * %i)),_
+lng2(1. + 1.4 * %i),logGamma(1. + 1.4 * %i),_
+lng2(1. + 1.4 * %i)-logGamma(1. + 1.4 * %i),_
+lng2(1. + 1.4 * %i)-(- 1.111864566426 - 0.204300724149 * %i)],_
+[1. + 1.5 * %i,- 1.234483051547 - 0.162939769480 * %i,_
+lng2(1. + 1.5 * %i),log(Gamma(1. + 1.5 * %i)),_
+lng2(1. + 1.5 * %i)-log(Gamma(1. + 1.5 * %i)),_
+lng2(1. + 1.5 * %i),logGamma(1. + 1.5 * %i),_
+lng2(1. + 1.5 * %i)-logGamma(1. + 1.5 * %i),_
+lng2(1. + 1.5 * %i)-(- 1.234483051547 - 0.162939769480 * %i)],_
+[1. + 1.6 * %i,- 1.359312248465 - 0.115468793589 * %i,_
+lng2(1. + 1.6 * %i),log(Gamma(1. + 1.6 * %i)),_
+lng2(1. + 1.6 * %i)-log(Gamma(1. + 1.6 * %i)),_
+lng2(1. + 1.6 * %i),logGamma(1. + 1.6 * %i),_
+lng2(1. + 1.6 * %i)-logGamma(1. + 1.6 * %i),_
+lng2(1. + 1.6 * %i)-(- 1.359312248465 - 0.115468793589 * %i)],_
+[1. + 1.7 * %i,- 1.486089612757 - 0.062198698329 * %i,_
+lng2(1. + 1.7 * %i),log(Gamma(1. + 1.7 * %i)),_
+lng2(1. + 1.7 * %i)-log(Gamma(1. + 1.7 * %i)),_
+lng2(1. + 1.7 * %i),logGamma(1. + 1.7 * %i),_
+lng2(1. + 1.7 * %i)-logGamma(1. + 1.7 * %i),_
+lng2(1. + 1.7 * %i)-(- 1.486089612757 - 0.062198698329 * %i)],_
+[1. + 1.8 * %i,- 1.614595396000 - 0.003416631477 * %i,_
+lng2(1. + 1.8 * %i),log(Gamma(1. + 1.8 * %i)),_
+lng2(1. + 1.8 * %i)-log(Gamma(1. + 1.8 * %i)),_
+lng2(1. + 1.8 * %i),logGamma(1. + 1.8 * %i),_
+lng2(1. + 1.8 * %i)-logGamma(1. + 1.8 * %i),_
+lng2(1. + 1.8 * %i)-(- 1.614595396000 - 0.003416631477 * %i)],_
+[1. + 1.9 * %i,- 1.744644276174 + 0.060612874295 * %i,_
+lng2(1. + 1.9 * %i),log(Gamma(1. + 1.9 * %i)),_
+lng2(1. + 1.9 * %i)-log(Gamma(1. + 1.9 * %i)),_
+lng2(1. + 1.9 * %i),logGamma(1. + 1.9 * %i),_
+lng2(1. + 1.9 * %i)-logGamma(1. + 1.9 * %i),_
+lng2(1. + 1.9 * %i)-(- 1.744644276174 + 0.060612874295 * %i)],_
+[1. + 2.0 * %i,- 1.876078786431 + 0.129646316310 * %i,_
+lng2(1. + 2.0 * %i),log(Gamma(1. + 2.0 * %i)),_
+lng2(1. + 2.0 * %i)-log(Gamma(1. + 2.0 * %i)),_
+lng2(1. + 2.0 * %i),logGamma(1. + 2.0 * %i),_
+lng2(1. + 2.0 * %i)-logGamma(1. + 2.0 * %i),_
+lng2(1. + 2.0 * %i)-(- 1.876078786431 + 0.129646316310 * %i)],_
+[1. + 2.1 * %i,- 2.008764150471 + 0.203459473833 * %i,_
+lng2(1. + 2.1 * %i),log(Gamma(1. + 2.1 * %i)),_
+lng2(1. + 2.1 * %i)-log(Gamma(1. + 2.1 * %i)),_
+lng2(1. + 2.1 * %i),logGamma(1. + 2.1 * %i),_
+lng2(1. + 2.1 * %i)-logGamma(1. + 2.1 * %i),_
+lng2(1. + 2.1 * %i)-(- 2.008764150471 + 0.203459473833 * %i)],_
+[1. + 2.2 * %i,- 2.142584209296 + 0.281845658426 * %i,_
+lng2(1. + 2.2 * %i),log(Gamma(1. + 2.2 * %i)),_
+lng2(1. + 2.2 * %i)-log(Gamma(1. + 2.2 * %i)),_
+lng2(1. + 2.2 * %i),logGamma(1. + 2.2 * %i),_
+lng2(1. + 2.2 * %i)-logGamma(1. + 2.2 * %i),_
+lng2(1. + 2.2 * %i)-(- 2.142584209296 + 0.281845658426 * %i)],_
+[1. + 2.3 * %i,- 2.277438192204 + 0.364614048950 * %i,_
+lng2(1. + 2.3 * %i),log(Gamma(1. + 2.3 * %i)),_
+lng2(1. + 2.3 * %i)-log(Gamma(1. + 2.3 * %i)),_
+lng2(1. + 2.3 * %i),logGamma(1. + 2.3 * %i),_
+lng2(1. + 2.3 * %i)-logGamma(1. + 2.3 * %i),_
+lng2(1. + 2.3 * %i)-(- 2.277438192204 + 0.364614048950 * %i)],_
+[1. + 2.4 * %i,- 2.413238141184 + 0.451588152441 * %i,_
+lng2(1. + 2.4 * %i),log(Gamma(1. + 2.4 * %i)),_
+lng2(1. + 2.4 * %i)-log(Gamma(1. + 2.4 * %i)),_
+lng2(1. + 2.4 * %i),logGamma(1. + 2.4 * %i),_
+lng2(1. + 2.4 * %i)-logGamma(1. + 2.4 * %i),_
+lng2(1. + 2.4 * %i)-(- 2.413238141184 + 0.451588152441 * %i)],_
+[1. + 2.5 * %i,- 2.549906842495 + 0.542604405852 * %i,_
+lng2(1. + 2.5 * %i),log(Gamma(1. + 2.5 * %i)),_
+lng2(1. + 2.5 * %i)-log(Gamma(1. + 2.5 * %i)),_
+lng2(1. + 2.5 * %i),logGamma(1. + 2.5 * %i),_
+lng2(1. + 2.5 * %i)-logGamma(1. + 2.5 * %i),_
+lng2(1. + 2.5 * %i)-(- 2.549906842495 + 0.542604405852 * %i)],_
+[1. + 2.6 * %i,- 2.687376153750 + 0.637510919046 * %i,_
+lng2(1. + 2.6 * %i),log(Gamma(1. + 2.6 * %i)),_
+lng2(1. + 2.6 * %i)-log(Gamma(1. + 2.6 * %i)),_
+lng2(1. + 2.6 * %i),logGamma(1. + 2.6 * %i),_
+lng2(1. + 2.6 * %i)-logGamma(1. + 2.6 * %i),_
+lng2(1. + 2.6 * %i)-(- 2.687376153750 + 0.637510919046 * %i)],_
+[1. + 2.7 * %i,- 2.825585641191 + 0.736166351679 * %i,_
+lng2(1. + 2.7 * %i),log(Gamma(1. + 2.7 * %i)),_
+lng2(1. + 2.7 * %i)-log(Gamma(1. + 2.7 * %i)),_
+lng2(1. + 2.7 * %i),logGamma(1. + 2.7 * %i),_
+lng2(1. + 2.7 * %i)-logGamma(1. + 2.7 * %i),_
+lng2(1. + 2.7 * %i)-(- 2.825585641191 + 0.736166351679 * %i)],_
+[1. + 2.8 * %i,- 2.964481461789 + 0.838438913096 * %i,_
+lng2(1. + 2.8 * %i),log(Gamma(1. + 2.8 * %i)),_
+lng2(1. + 2.8 * %i)-log(Gamma(1. + 2.8 * %i)),_
+lng2(1. + 2.8 * %i),logGamma(1. + 2.8 * %i),_
+lng2(1. + 2.8 * %i)-logGamma(1. + 2.8 * %i),_
+lng2(1. + 2.8 * %i)-(- 2.964481461789 + 0.838438913096 * %i)],_
+[1. + 2.9 * %i,- 3.104015439901 + 0.944205473039 * %i,_
+lng2(1. + 2.9 * %i),log(Gamma(1. + 2.9 * %i)),_
+lng2(1. + 2.9 * %i)-log(Gamma(1. + 2.9 * %i)),_
+lng2(1. + 2.9 * %i),logGamma(1. + 2.9 * %i),_
+lng2(1. + 2.9 * %i)-logGamma(1. + 2.9 * %i),_
+lng2(1. + 2.9 * %i)-(- 3.104015439901 + 0.944205473039 * %i)],_
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+lng2(1. + 7.0 * %i)-(- 9.103680679832 + 7.394856298436 * %i)],_
+[1. + 7.1 * %i,- 9.253667995015 + 7.590326235184 * %i,_
+lng2(1. + 7.1 * %i),log(Gamma(1. + 7.1 * %i)),_
+lng2(1. + 7.1 * %i)-log(Gamma(1. + 7.1 * %i)),_
+lng2(1. + 7.1 * %i),logGamma(1. + 7.1 * %i),_
+lng2(1. + 7.1 * %i)-logGamma(1. + 7.1 * %i),_
+lng2(1. + 7.1 * %i)-(- 9.253667995015 + 7.590326235184 * %i)],_
+[1. + 7.2 * %i,- 9.403754506708 + 7.787199992877 * %i,_
+lng2(1. + 7.2 * %i),log(Gamma(1. + 7.2 * %i)),_
+lng2(1. + 7.2 * %i)-log(Gamma(1. + 7.2 * %i)),_
+lng2(1. + 7.2 * %i),logGamma(1. + 7.2 * %i),_
+lng2(1. + 7.2 * %i)-logGamma(1. + 7.2 * %i),_
+lng2(1. + 7.2 * %i)-(- 9.403754506708 + 7.787199992877 * %i)],_
+[1. + 7.3 * %i,- 9.553937478321 + 7.985458200468 * %i,_
+lng2(1. + 7.3 * %i),log(Gamma(1. + 7.3 * %i)),_
+lng2(1. + 7.3 * %i)-log(Gamma(1. + 7.3 * %i)),_
+lng2(1. + 7.3 * %i),logGamma(1. + 7.3 * %i),_
+lng2(1. + 7.3 * %i)-logGamma(1. + 7.3 * %i),_
+lng2(1. + 7.3 * %i)-(- 9.553937478321 + 7.985458200468 * %i)],_
+[1. + 7.4 * %i,- 9.704214284972 + 8.185082012503 * %i,_
+lng2(1. + 7.4 * %i),log(Gamma(1. + 7.4 * %i)),_
+lng2(1. + 7.4 * %i)-log(Gamma(1. + 7.4 * %i)),_
+lng2(1. + 7.4 * %i),logGamma(1. + 7.4 * %i),_
+lng2(1. + 7.4 * %i)-logGamma(1. + 7.4 * %i),_
+lng2(1. + 7.4 * %i)-(- 9.704214284972 + 8.185082012503 * %i)],_
+[1. + 7.5 * %i,- 9.854582407486 + 8.386053088089 * %i,_
+lng2(1. + 7.5 * %i),log(Gamma(1. + 7.5 * %i)),_
+lng2(1. + 7.5 * %i)-log(Gamma(1. + 7.5 * %i)),_
+lng2(1. + 7.5 * %i),logGamma(1. + 7.5 * %i),_
+lng2(1. + 7.5 * %i)-logGamma(1. + 7.5 * %i),_
+lng2(1. + 7.5 * %i)-(- 9.854582407486 + 8.386053088089 * %i)],_
+[1. + 7.6 * %i,- 10.005039426790 + 8.588353570962 * %i,_
+lng2(1. + 7.6 * %i),log(Gamma(1. + 7.6 * %i)),_
+lng2(1. + 7.6 * %i)-log(Gamma(1. + 7.6 * %i)),_
+lng2(1. + 7.6 * %i),logGamma(1. + 7.6 * %i),_
+lng2(1. + 7.6 * %i)-logGamma(1. + 7.6 * %i),_
+lng2(1. + 7.6 * %i)-(- 10.005039426790 + 8.588353570962 * %i)],_
+[1. + 7.7 * %i,- 10.155583018686 + 8.791966070587 * %i,_
+lng2(1. + 7.7 * %i),log(Gamma(1. + 7.7 * %i)),_
+lng2(1. + 7.7 * %i)-log(Gamma(1. + 7.7 * %i)),_
+lng2(1. + 7.7 * %i),logGamma(1. + 7.7 * %i),_
+lng2(1. + 7.7 * %i)-logGamma(1. + 7.7 * %i),_
+lng2(1. + 7.7 * %i)-(- 10.155583018686 + 8.791966070587 * %i)],_
+[1. + 7.8 * %i,- 10.306210948948 + 8.996873644229 * %i,_
+lng2(1. + 7.8 * %i),log(Gamma(1. + 7.8 * %i)),_
+lng2(1. + 7.8 * %i)-log(Gamma(1. + 7.8 * %i)),_
+lng2(1. + 7.8 * %i),logGamma(1. + 7.8 * %i),_
+lng2(1. + 7.8 * %i)-logGamma(1. + 7.8 * %i),_
+lng2(1. + 7.8 * %i)-(- 10.306210948948 + 8.996873644229 * %i)],_
+[1. + 7.9 * %i,- 10.456921068739 + 9.203059779925 * %i,_
+lng2(1. + 7.9 * %i),log(Gamma(1. + 7.9 * %i)),_
+lng2(1. + 7.9 * %i)-log(Gamma(1. + 7.9 * %i)),_
+lng2(1. + 7.9 * %i),logGamma(1. + 7.9 * %i),_
+lng2(1. + 7.9 * %i)-logGamma(1. + 7.9 * %i),_
+lng2(1. + 7.9 * %i)-(- 10.456921068739 + 9.203059779925 * %i)],_
+[1. + 8.0 * %i,- 10.607711310315 + 9.410508380312 * %i,_
+lng2(1. + 8.0 * %i),log(Gamma(1. + 8.0 * %i)),_
+lng2(1. + 8.0 * %i)-log(Gamma(1. + 8.0 * %i)),_
+lng2(1. + 8.0 * %i),logGamma(1. + 8.0 * %i),_
+lng2(1. + 8.0 * %i)-logGamma(1. + 8.0 * %i),_
+lng2(1. + 8.0 * %i)-(- 10.607711310315 + 9.410508380312 * %i)],_
+[1. + 8.1 * %i,- 10.758579682995 + 9.619203747242 * %i,_
+lng2(1. + 8.1 * %i),log(Gamma(1. + 8.1 * %i)),_
+lng2(1. + 8.1 * %i)-log(Gamma(1. + 8.1 * %i)),_
+lng2(1. + 8.1 * %i),logGamma(1. + 8.1 * %i),_
+lng2(1. + 8.1 * %i)-logGamma(1. + 8.1 * %i),_
+lng2(1. + 8.1 * %i)-(- 10.758579682995 + 9.619203747242 * %i)],_
+[1. + 8.2 * %i,- 10.909524269378 + 9.829130567162 * %i,_
+lng2(1. + 8.2 * %i),log(Gamma(1. + 8.2 * %i)),_
+lng2(1. + 8.2 * %i)-log(Gamma(1. + 8.2 * %i)),_
+lng2(1. + 8.2 * %i),logGamma(1. + 8.2 * %i),_
+lng2(1. + 8.2 * %i)-logGamma(1. + 8.2 * %i),_
+lng2(1. + 8.2 * %i)-(- 10.909524269378 + 9.829130567162 * %i)],_
+[1. + 8.3 * %i,- 11.060543221792 + 10.040273897180 * %i,_
+lng2(1. + 8.3 * %i),log(Gamma(1. + 8.3 * %i)),_
+lng2(1. + 8.3 * %i)-log(Gamma(1. + 8.3 * %i)),_
+lng2(1. + 8.3 * %i),logGamma(1. + 8.3 * %i),_
+lng2(1. + 8.3 * %i)-logGamma(1. + 8.3 * %i),_
+lng2(1. + 8.3 * %i)-(- 11.060543221792 + 10.040273897180 * %i)],_
+[1. + 8.4 * %i,- 11.211634758948 + 10.252619151809 * %i,_
+lng2(1. + 8.4 * %i),log(Gamma(1. + 8.4 * %i)),_
+lng2(1. + 8.4 * %i)-log(Gamma(1. + 8.4 * %i)),_
+lng2(1. + 8.4 * %i),logGamma(1. + 8.4 * %i),_
+lng2(1. + 8.4 * %i)-logGamma(1. + 8.4 * %i),_
+lng2(1. + 8.4 * %i)-(- 11.211634758948 + 10.252619151809 * %i)],_
+[1. + 8.5 * %i,- 11.362797162804 + 10.466152090324 * %i,_
+lng2(1. + 8.5 * %i),log(Gamma(1. + 8.5 * %i)),_
+lng2(1. + 8.5 * %i)-log(Gamma(1. + 8.5 * %i)),_
+lng2(1. + 8.5 * %i),logGamma(1. + 8.5 * %i),_
+lng2(1. + 8.5 * %i)-logGamma(1. + 8.5 * %i),_
+lng2(1. + 8.5 * %i)-(- 11.362797162804 + 10.466152090324 * %i)],_
+[1. + 8.6 * %i,- 11.514028775602 + 10.680858804712 * %i,_
+lng2(1. + 8.6 * %i),log(Gamma(1. + 8.6 * %i)),_
+lng2(1. + 8.6 * %i)-log(Gamma(1. + 8.6 * %i)),_
+lng2(1. + 8.6 * %i),logGamma(1. + 8.6 * %i),_
+lng2(1. + 8.6 * %i)-logGamma(1. + 8.6 * %i),_
+lng2(1. + 8.6 * %i)-(- 11.514028775602 + 10.680858804712 * %i)],_
+[1. + 8.7 * %i,- 11.665327997081 + 10.896725708177 * %i,_
+lng2(1. + 8.7 * %i),log(Gamma(1. + 8.7 * %i)),_
+lng2(1. + 8.7 * %i)-log(Gamma(1. + 8.7 * %i)),_
+lng2(1. + 8.7 * %i),logGamma(1. + 8.7 * %i),_
+lng2(1. + 8.7 * %i)-logGamma(1. + 8.7 * %i),_
+lng2(1. + 8.7 * %i)-(- 11.665327997081 + 10.896725708177 * %i)],_
+[1. + 8.8 * %i,- 11.816693281848 + 11.113739524157 * %i,_
+lng2(1. + 8.8 * %i),log(Gamma(1. + 8.8 * %i)),_
+lng2(1. + 8.8 * %i)-log(Gamma(1. + 8.8 * %i)),_
+lng2(1. + 8.8 * %i),logGamma(1. + 8.8 * %i),_
+lng2(1. + 8.8 * %i)-logGamma(1. + 8.8 * %i),_
+lng2(1. + 8.8 * %i)-(- 11.816693281848 + 11.113739524157 * %i)],_
+[1. + 8.9 * %i,- 11.968123136901 + 11.331887275853 * %i,_
+lng2(1. + 8.9 * %i),log(Gamma(1. + 8.9 * %i)),_
+lng2(1. + 8.9 * %i)-log(Gamma(1. + 8.9 * %i)),_
+lng2(1. + 8.9 * %i),logGamma(1. + 8.9 * %i),_
+lng2(1. + 8.9 * %i)-logGamma(1. + 8.9 * %i),_
+lng2(1. + 8.9 * %i)-(- 11.968123136901 + 11.331887275853 * %i)],_
+[1. + 9.0 * %i,- 12.119616119281 + 11.551156276202 * %i,_
+lng2(1. + 9.0 * %i),log(Gamma(1. + 9.0 * %i)),_
+lng2(1. + 9.0 * %i)-log(Gamma(1. + 9.0 * %i)),_
+lng2(1. + 9.0 * %i),logGamma(1. + 9.0 * %i),_
+lng2(1. + 9.0 * %i)-logGamma(1. + 9.0 * %i),_
+lng2(1. + 9.0 * %i)-(- 12.119616119281 + 11.551156276202 * %i)],_
+[1. + 9.1 * %i,- 12.271170833867 + 11.771534118309 * %i,_
+lng2(1. + 9.1 * %i),log(Gamma(1. + 9.1 * %i)),_
+lng2(1. + 9.1 * %i)-log(Gamma(1. + 9.1 * %i)),_
+lng2(1. + 9.1 * %i),logGamma(1. + 9.1 * %i),_
+lng2(1. + 9.1 * %i)-logGamma(1. + 9.1 * %i),_
+lng2(1. + 9.1 * %i)-(- 12.271170833867 + 11.771534118309 * %i)],_
+[1. + 9.2 * %i,- 12.422785931281 + 11.993008666285 * %i,_
+lng2(1. + 9.2 * %i),log(Gamma(1. + 9.2 * %i)),_
+lng2(1. + 9.2 * %i)-log(Gamma(1. + 9.2 * %i)),_
+lng2(1. + 9.2 * %i),logGamma(1. + 9.2 * %i),_
+lng2(1. + 9.2 * %i)-logGamma(1. + 9.2 * %i),_
+lng2(1. + 9.2 * %i)-(- 12.422785931281 + 11.993008666285 * %i)],_
+[1. + 9.3 * %i,- 12.574460105908 + 12.215568046479 * %i,_
+lng2(1. + 9.3 * %i),log(Gamma(1. + 9.3 * %i)),_
+lng2(1. + 9.3 * %i)-log(Gamma(1. + 9.3 * %i)),_
+lng2(1. + 9.3 * %i),logGamma(1. + 9.3 * %i),_
+lng2(1. + 9.3 * %i)-logGamma(1. + 9.3 * %i),_
+lng2(1. + 9.3 * %i)-(- 12.574460105908 + 12.215568046479 * %i)],_
+[1. + 9.4 * %i,- 12.726192094029 + 12.439200639090 * %i,_
+lng2(1. + 9.4 * %i),log(Gamma(1. + 9.4 * %i)),_
+lng2(1. + 9.4 * %i)-log(Gamma(1. + 9.4 * %i)),_
+lng2(1. + 9.4 * %i),logGamma(1. + 9.4 * %i),_
+lng2(1. + 9.4 * %i)-logGamma(1. + 9.4 * %i),_
+lng2(1. + 9.4 * %i)-(- 12.726192094029 + 12.439200639090 * %i)],_
+[1. + 9.5 * %i,- 12.877980672044 + 12.663895070128 * %i,_
+lng2(1. + 9.5 * %i),log(Gamma(1. + 9.5 * %i)),_
+lng2(1. + 9.5 * %i)-log(Gamma(1. + 9.5 * %i)),_
+lng2(1. + 9.5 * %i),logGamma(1. + 9.5 * %i),_
+lng2(1. + 9.5 * %i)-logGamma(1. + 9.5 * %i),_
+lng2(1. + 9.5 * %i)-(- 12.877980672044 + 12.663895070128 * %i)],_
+[1. + 9.6 * %i,- 13.029824654789 + 12.889640203708 * %i,_
+lng2(1. + 9.6 * %i),log(Gamma(1. + 9.6 * %i)),_
+lng2(1. + 9.6 * %i)-log(Gamma(1. + 9.6 * %i)),_
+lng2(1. + 9.6 * %i),logGamma(1. + 9.6 * %i),_
+lng2(1. + 9.6 * %i)-logGamma(1. + 9.6 * %i),_
+lng2(1. + 9.6 * %i)-(- 13.029824654789 + 12.889640203708 * %i)],_
+[1. + 9.7 * %i,- 13.181722893951 + 13.116425134666 * %i,_
+lng2(1. + 9.7 * %i),log(Gamma(1. + 9.7 * %i)),_
+lng2(1. + 9.7 * %i)-log(Gamma(1. + 9.7 * %i)),_
+lng2(1. + 9.7 * %i),logGamma(1. + 9.7 * %i),_
+lng2(1. + 9.7 * %i)-logGamma(1. + 9.7 * %i),_
+lng2(1. + 9.7 * %i)-(- 13.181722893951 + 13.116425134666 * %i)],_
+[1. + 9.8 * %i,- 13.333674276547 + 13.344239181477 * %i,_
+lng2(1. + 9.8 * %i),log(Gamma(1. + 9.8 * %i)),_
+lng2(1. + 9.8 * %i)-log(Gamma(1. + 9.8 * %i)),_
+lng2(1. + 9.8 * %i),logGamma(1. + 9.8 * %i),_
+lng2(1. + 9.8 * %i)-logGamma(1. + 9.8 * %i),_
+lng2(1. + 9.8 * %i)-(- 13.333674276547 + 13.344239181477 * %i)],_
+[1. + 9.9 * %i,- 13.485677723495 + 13.573071879455 * %i,_
+lng2(1. + 9.9 * %i),log(Gamma(1. + 9.9 * %i)),_
+lng2(1. + 9.9 * %i)-log(Gamma(1. + 9.9 * %i)),_
+lng2(1. + 9.9 * %i),logGamma(1. + 9.9 * %i),_
+lng2(1. + 9.9 * %i)-logGamma(1. + 9.9 * %i),_
+lng2(1. + 9.9 * %i)-(- 13.485677723495 + 13.573071879455 * %i)],_
+[1. + 10.0 * %i,- 13.637732188247 + 13.802912974230 * %i,_
+lng2(1. + 10.0 * %i),log(Gamma(1. + 10.0 * %i)),_
+lng2(1. + 10.0 * %i)-log(Gamma(1. + 10.0 * %i)),_
+lng2(1. + 10.0 * %i),logGamma(1. + 10.0 * %i),_
+lng2(1. + 10.0 * %i)-logGamma(1. + 10.0 * %i),_
+lng2(1. + 10.0 * %i)-(- 13.637732188247 + 13.802912974230 * %i)]]
+--R
+--R Compiling function lng2 with type Complex DoubleFloat -> Complex
+--R DoubleFloat
+--R
+--R (12)
+--R [[1.,0.,0.,0.],
+--R
+--R [1. + 0.10000000000000001 %i,
+--R - 8.1977805649999999E-3 - 5.7322940417E-2 %i,
+--R - 8.1977805654074309E-3 - 5.732294041672345E-2 %i,
+--R - 8.1977805654051359E-3 - 5.7322940416719675E-2 %i,
+--R - 2.2950391587173158E-15 - 3.7747582837255322E-15 %i,
+--R - 8.1977805654074309E-3 - 5.732294041672345E-2 %i,
+--R - 8.1977805654052105E-3 - 5.7322940416719675E-2 %i,
+--R - 2.2204460492503131E-15 - 3.7747582837255322E-15 %i,
+--R - 4.0743103335572073E-13 + 2.7654961654022259E-13 %i]
+--R ,
+--R
+--R [1. + 0.20000000000000001 %i, - 3.2476292317999998E-2 - 0.112302222644
%i,
+--R - 3.2476292318133204E-2 - 0.11230222264419082 %i,
+--R - 3.2476292318128805E-2 - 0.11230222264418371 %i,
+--R - 4.3992587350771828E-15 - 7.1054273576010019E-15 %i,
+--R - 3.2476292318133204E-2 - 0.11230222264419082 %i,
+--R - 3.2476292318128763E-2 - 0.11230222264418371 %i,
+--R - 4.4408920985006262E-15 - 7.1054273576010019E-15 %i,
+--R - 1.3320594627330706E-13 - 1.9081958235744878E-13 %i]
+--R ,
+--R
+--R [1. + 0.29999999999999999 %i, - 7.1946250899999994E-2 - 0.162820672168
%i,
+--R - 7.1946250899646902E-2 - 0.16282067216786528 %i,
+--R - 7.1946250899640213E-2 - 0.16282067216785573 %i,
+--R - 6.6890937233665682E-15 - 9.5479180117763462E-15 %i,
+--R - 7.1946250899646902E-2 - 0.16282067216786528 %i,
+--R - 7.1946250899640241E-2 - 0.16282067216785573 %i,
+--R - 6.6613381477509392E-15 - 9.5479180117763462E-15 %i,
+--R 3.5309255519422322E-13 + 1.3472556403826275E-13 %i]
+--R ,
+--R
+--R [1. + 0.40000000000000002 %i, - 0.125289374821 - 0.20715582631599999
%i,
+--R - 0.12528937482072333 - 0.20715582631567919 %i,
+--R - 0.12528937482070648 - 0.20715582631566853 %i,
+--R - 1.6847634398686751E-14 - 1.0658141036401503E-14 %i,
+--R - 0.12528937482072333 - 0.20715582631567919 %i,
+--R - 0.12528937482070646 - 0.20715582631566853 %i,
+--R - 1.6875389974302379E-14 - 1.0658141036401503E-14 %i,
+--R 2.7666757773658901E-13 + 3.2079894296543898E-13 %i]
+--R ,
+--R
+--R [1. + 0.5 %i, - 0.19094549918699999 - 0.244058298905 %i,
+--R - 0.19094549918680226 - 0.24405829890543784 %i,
+--R - 0.19094549918678008 - 0.24405829890542749 %i,
+--R - 2.2176704916887502E-14 - 1.0352829704629585E-14 %i,
+--R - 0.19094549918680226 - 0.24405829890543784 %i,
+--R - 0.19094549918678005 - 0.24405829890542752 %i,
+--R - 2.2204460492503131E-14 - 1.0325074129013956E-14 %i,
+--R 1.9773072068574038E-13 - 4.3784420533654611E-13 %i]
+--R ,
+--R
+--R [1. + 0.59999999999999998 %i,
+--R - 0.26729006821399998 - 0.27274381049099999 %i,
+--R - 0.26729006821416545 - 0.27274381049105989 %i,
+--R - 0.2672900682141322 - 0.27274381049105378 %i,
+--R - 3.3251179587523438E-14 - 6.106226635438361E-15 %i,
+--R - 0.26729006821416545 - 0.27274381049105989 %i,
+--R - 0.26729006821413215 - 0.27274381049105378 %i,
+--R - 3.3306690738754696E-14 - 6.106226635438361E-15 %i,
+--R - 1.6547874182037958E-13 - 5.9896532178527195E-14 %i]
+--R ,
+--R
+--R [1. + 0.69999999999999996 %i,
+--R - 0.35276869086000001 - 0.29282635118700001 %i,
+--R - 0.35276869085965368 - 0.29282635118686051 %i,
+--R - 0.3527686908596116 - 0.29282635118686201 %i,
+--R - 4.2077452633293433E-14 + 1.4988010832439613E-15 %i,
+--R - 0.35276869085965368 - 0.29282635118686051 %i,
+--R - 0.35276869085961149 - 0.29282635118686196 %i,
+--R - 4.2188474935755949E-14 + 1.4432899320127035E-15 %i,
+--R 3.4633407253181758E-13 + 1.3949952304415092E-13 %i]
+--R ,
+--R
+--R [1. + 0.80000000000000004 %i, - 0.445978783549 - 0.30422560297599999
%i,
+--R - 0.4459787835488167 - 0.30422560297617007 %i,
+--R - 0.4459787835487648 - 0.30422560297618323 %i,
+--R - 5.1902926401226068E-14 + 1.3156142841808105E-14 %i,
+--R - 0.4459787835488167 - 0.30422560297617007 %i,
+--R - 0.44597878354876475 - 0.30422560297618317 %i,
+--R - 5.1958437552457326E-14 + 1.3100631690576847E-14 %i,
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+--R ,
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%i,
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+--R ,
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%i,
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%i,
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+--R ,
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%i,
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+--R ,
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+--R ,
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+--R [1. + 8. %i, - 10.607711310315 + 9.4105083803120007 %i,
+--R - 10.607711310271373 + 9.4105083803211791 %i,
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+--R ,
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+--R [1. + 8.0999999999999996 %i, - 10.758579682995 + 9.6192037472420004 %i,
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+--R ,
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+--R [1. + 8.1999999999999993 %i, - 10.909524269378 + 9.8291305671620002 %i,
+--R - 10.909524269331923 + 9.8291305671662812 %i,
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+--R 4.6076920057203097E-11 + 4.2810199829546036E-12 %i]
+--R ,
+--R
+--R [1. + 8.3000000000000007 %i, - 11.060543221792001 + 10.040273897180001
%i,
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+--R - 11.060543221743844 + 10.040273897181736 %i,
+--R - 11.060543221791693 + 10.040273897179832 %i,
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+--R 4.815703391614079E-11 + 1.7355006320940447E-12 %i]
+--R ,
+--R
+--R [1. + 8.4000000000000004 %i, - 11.211634758948 + 10.252619151809 %i,
+--R - 11.211634758898732 + 10.252619151807744 %i,
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+--R - 11.211634758947826 + 10.252619151808615 %i,
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+--R 4.9267256940765947E-11 - 1.2558842854559771E-12 %i]
+--R ,
+--R
+--R [1. + 8.5 %i, - 11.362797162804 + 10.466152090324 %i,
+--R - 11.36279716275364 + 10.466152090319857 %i,
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+--R - 11.36279716275364 + 10.466152090319857 %i,
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+--R 5.0359716396997101E-11 - 4.1424641494813841E-12 %i]
+--R ,
+--R
+--R [1. + 8.5999999999999996 %i, - 11.514028775602 + 10.680858804712001 %i,
+--R - 11.514028775550621 + 10.680858804705615 %i,
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+--R - 11.514028775550621 + 10.680858804705615 %i,
+--R - 11.514028775601707 + 10.680858804712296 %i,
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+--R 5.1379345222812844E-11 - 6.3860028376439004E-12 %i]
+--R ,
+--R
+--R [1. + 8.6999999999999993 %i, - 11.665327997081 + 10.896725708177 %i,
+--R - 11.665327997028827 + 10.89672570816688 %i,
+--R - 11.665327997080658 - 1.6696449061826013 %i,
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+--R - 11.665327997028827 + 10.89672570816688 %i,
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+--R 5.2173376730024756E-11 - 1.0119904914063227E-11 %i]
+--R ,
+--R
+--R [1. + 8.8000000000000007 %i, - 11.816693281848 + 11.113739524156999 %i,
+--R - 11.816693281795935 + 11.113739524144647 %i,
+--R - 11.816693281848337 - 1.452631090201765 %i,
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+--R - 11.816693281795935 + 11.113739524144647 %i,
+--R - 11.816693281848337 + 11.113739524157408 %i,
+--R 5.2402526762307389E-11 - 1.2761347534251399E-11 %i,
+--R 5.2065018962821341E-11 - 1.2352785461189342E-11 %i]
+--R ,
+--R
+--R [1. + 8.9000000000000004 %i, - 11.968123136900999 + 11.331887275852999
%i,
+--R - 11.968123136848055 + 11.331887275837042 %i,
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+--R - 11.968123136900861 + 11.331887275852912 %i,
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+--R 5.2944315598324465E-11 - 1.595701348833245E-11 %i]
+--R ,
+--R
+--R [1. + 9. %i, - 12.119616119281 + 11.551156276202001 %i,
+--R - 12.119616119228253 + 11.551156276183157 %i,
+--R - 12.119616119281286 - 1.0152143381569982 %i,
+--R 5.3033133440294478E-11 + 12.566370614340155 %i,
+--R - 12.119616119228253 + 11.551156276183157 %i,
+--R - 12.119616119281286 + 11.551156276202175 %i,
+--R 5.3033133440294478E-11 - 1.9017676322619081E-11 %i,
+--R 5.2747139989151037E-11 - 1.8843593352357857E-11 %i]
+--R ,
+--R
+--R [1. + 9.0999999999999996 %i, - 12.271170833867 + 11.771534118309001 %i,
+--R - 12.271170833814395 + 11.77153411828726 %i,
+--R - 12.271170833867483 - 0.79483649604973161 %i,
+--R 5.3088200502315885E-11 + 12.566370614336991 %i,
+--R - 12.271170833814395 + 11.77153411828726 %i,
+--R - 12.271170833867483 + 11.771534118309441 %i,
+--R 5.3088200502315885E-11 - 2.2181367853590928E-11 %i,
+--R 5.2605031441999017E-11 - 2.1740831357419665E-11 %i]
+--R ,
+--R
+--R [1. + 9.1999999999999993 %i, - 12.422785931281 + 11.993008666285 %i,
+--R - 12.422785931227907 + 11.993008666259376 %i,
+--R - 12.422785931280877 - 0.5733619480744393 %i,
+--R 5.2970960950915469E-11 + 12.566370614333815 %i,
+--R - 12.422785931227907 + 11.993008666259376 %i,
+--R - 12.422785931280877 + 11.993008666284734 %i,
+--R 5.2970960950915469E-11 - 2.5357493882438575E-11 %i,
+--R 5.3093529572834086E-11 - 2.5623947408348613E-11 %i]
+--R ,
+--R
+--R [1. + 9.3000000000000007 %i, - 12.574460105908001 + 12.215568046479 %i,
+--R - 12.574460105855573 + 12.215568046450077 %i,
+--R - 12.574460105908262 - 0.35080256788055886 %i,
+--R 5.2688520213450829E-11 + 12.566370614330635 %i,
+--R - 12.574460105855573 + 12.215568046450077 %i,
+--R - 12.574460105908262 + 12.215568046478614 %i,
+--R 5.2688520213450829E-11 - 2.8537172624965024E-11 %i,
+--R 5.2427395758058992E-11 - 2.8922642059114878E-11 %i]
+--R ,
+--R
+--R [1. + 9.4000000000000004 %i, - 12.726192094029001 + 12.43920063909 %i,
+--R - 12.726192093977144 + 12.43920063905834 %i,
+--R - 12.726192094029377 - 0.12716997526913024 %i,
+--R 5.2233772862564365E-11 + 12.56637061432747 %i,
+--R - 12.726192093977144 + 12.43920063905834 %i,
+--R - 12.726192094029377 + 12.439200639090043 %i,
+--R 5.2233772862564365E-11 - 3.170264051277627E-11 %i,
+--R 5.1857185212611512E-11 - 3.1660007948630664E-11 %i]
+--R ,
+--R
+--R [1. + 9.5 %i, - 12.877980672044 + 12.663895070128 %i,
+--R - 12.877980671991985 + 12.663895070093062 %i,
+--R - 12.877980672043599 + 9.7524455768741289E-2 %i,
+--R 5.1613824325613678E-11 + 12.56637061432432 %i,
+--R - 12.877980671991985 + 12.663895070093062 %i,
+--R - 12.877980672043599 + 12.663895070127914 %i,
+--R 5.1613824325613678E-11 - 3.4852121189032914E-11 %i,
+--R 5.2015280971318134E-11 - 3.4937386317324126E-11 %i]
+--R ,
+--R
+--R [1. + 9.5999999999999996 %i, - 13.029824654789 + 12.889640203708 %i,
+--R - 13.029824654738606 + 12.889640203669721 %i,
+--R - 13.02982465478944 + 0.32326958934851951 %i,
+--R 5.0834003673116968E-11 + 12.566370614321201 %i,
+--R - 13.029824654738606 + 12.889640203669721 %i,
+--R - 13.02982465478944 + 12.889640203707692 %i,
+--R 5.0834003673116968E-11 - 3.7971403799019754E-11 %i,
+--R 5.0393467176945705E-11 - 3.8278713532235997E-11 %i]
+--R ,
+--R
+--R [1. + 9.6999999999999993 %i, - 13.181722893950999 + 13.116425134666001
%i,
+--R - 13.181722893901259 + 13.116425134624947 %i,
+--R - 13.181722893951155 + 0.55005452030683832 %i,
+--R 4.9896087261913635E-11 + 12.566370614318108 %i,
+--R - 13.181722893901259 + 13.116425134624947 %i,
+--R - 13.181722893951155 + 13.116425134666011 %i,
+--R 4.9896087261913635E-11 - 4.106404105641559E-11 %i,
+--R 4.9739767860046413E-11 - 4.1053382915379188E-11 %i]
+--R ,
+--R
+--R [1. + 9.8000000000000007 %i, - 13.333674276547001 + 13.344239181477 %i,
+--R - 13.33367427649825 + 13.344239181432879 %i,
+--R - 13.333674276547052 + 0.77786856711780805 %i,
+--R 4.8801851448843081E-11 + 12.566370614315071 %i,
+--R - 13.33367427649825 + 13.344239181432879 %i,
+--R - 13.333674276547052 + 13.344239181476981 %i,
+--R 4.8801851448843081E-11 - 4.4101611251790018E-11 %i,
+--R 4.8750337100500474E-11 - 4.4121151177023421E-11 %i]
+--R ,
+--R
+--R [1. + 9.9000000000000004 %i, - 13.485677723495 + 13.573071879455 %i,
+--R - 13.485677723446969 + 13.573071879407928 %i,
+--R - 13.485677723494533 + 1.0067012650958465 %i,
+--R 4.7563730731781106E-11 + 12.566370614312081 %i,
+--R - 13.485677723446969 + 13.573071879407928 %i,
+--R - 13.485677723494533 + 13.57307187945502 %i,
+--R 4.7563730731781106E-11 - 4.709121981250064E-11 %i,
+--R 4.8030912580543372E-11 - 4.7071679887267237E-11 %i]
+--R ,
+--R
+--R [1. + 10. %i, - 13.637732188247 + 13.802912974230001 %i,
+--R - 13.637732188201092 + 13.802912974179876 %i,
+--R - 13.637732188247268 + 1.2365423598707301 %i,
+--R 4.6176396040209511E-11 + 12.566370614309145 %i,
+--R - 13.637732188201092 + 13.802912974179876 %i,
+--R - 13.637732188247268 + 13.802912974229903 %i,
+--R 4.6176396040209511E-11 - 5.0027537668029254E-11 %i,
+--R 4.5908166157460073E-11 - 5.0125237294196268E-11 %i]
+--R ]
+--R Type: List List Complex
DoubleFloat
+--E 12
)spool
)lisp (bye)
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- [Axiom-developer] 20080222.01.tpd.patch (7099 logGamma vs log(Gamma) ),
daly <=