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Re: [Axiom-developer] 20080221.01.tpd.patch (7099: complex gamma functio

From: Waldek Hebisch
Subject: Re: [Axiom-developer] 20080221.01.tpd.patch (7099: complex gamma function investigation)
Date: Fri, 22 Feb 2008 00:22:21 +0100 (CET)

> 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.

(11) -> Gamma(1. + 4.5*%i)

   (11)  - 0.004501804477919395 + 4.8078797963506284e-4 %i
                                                    Type: Complex DoubleFloat
(12) -> Gamma(1. + 4.6*%i)

   (12)  - 0.0039079873004091254 - 1.7801308638883733e-4 %i
                                                    Type: Complex DoubleFloat

So, value of Gamma crossed brunch cut of logarithm and we have jump
in numeric value by -2Pi.  However, the formulas in texbooks (including
Abramowitz and Stegun) do not use numeric logarithm: log(Gamma(z))
in texbooks is a holomrphic function for Re z > 0, in particular
texbook log(Gamma(z)) is continouos, while numeric one have jumps.

                              Waldek Hebisch

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