Re: Complex path integral wrong

*To*: mathgroup at smc.vnet.net*Subject*: [mg131371] Re: Complex path integral wrong*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Tue, 2 Jul 2013 00:44:22 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-outx@smc.vnet.net*Delivered-to*: mathgroup-newsendx@smc.vnet.net*References*: <kqomj6$p76$1@smc.vnet.net> <20130701095219.0C6426ABC@smc.vnet.net>

Hmm.. Documentation Center page on Hypergeometric2F1 says: "Hypergeometric2F1[a, b, c, z] has a branch cut discontinuity in the complex z plane running from 1 to Infinity." Page http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/04/02/ says: "[Hypergeometric2F1[a, b, c, z]] is an analytical function of a, b, c, and z which is defined on C^4" Hence for fixed a, b, and c, that implies it is analytic as a function of z on the complex plane. Which is it? On Jul 1, 2013, at 5:52 AM, Kevin J. McCann <kjm at kevinmccann.com> wrote: > A followup on my earlier reply. > > If I compare the analytical and numerical results from Mathematica, the > discrepancy occurs on the first leg of the integration > (1+I -> 1+I R). It is clear that the numerical integration is correct. > > Could this be some kind of branch cut issue with the hypergeometric > function evaluation? > > Kevin > > On 6/30/2013 3:26 AM, Dr. Wolfgang Hintze wrote: >> I suspect this is a bug >> In[361]:= $Version >> Out[361]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" >> >> The follwing path integral comes out wrong: >> >> R = 3 \[Pi] ; >> Integrate[Exp[I s]/( >> Exp[s] - 1 ), {s, 1 + I, 1 + I R, -1 + I R, -1 + I, 1 + I}] // FullSimplify >> >> Out[351]= 0 >> >> It should have the value >> >> In[356]:= (2 \[Pi] I) Residue[Exp[I s]/(Exp[s] - 1 ), {s, 2 \[Pi] I}] >> >> Out[356]= (2 \[Pi] I) E^(-2 \[Pi]) >> >> Without applying FullSimplify the result of the integration is >> >> In[357]:= R = 3*Pi; >> Integrate[ >> Exp[I*s]/(Exp[s] - 1), {s, 1 + I, 1 + I*R, -1 + I*R, -1 + I, 1 + I}] >> >> Out[358]= >> I*E^((-1 - I) - 3*Pi)*((-E)*Hypergeometric2F1[I, 1, 1 + I, -(1/E)] + >> E^(3*Pi)*Hypergeometric2F1[I, 1, 1 + I, E^(-1 + I)]) + >> I*E^(-I - 3*Pi)*(Hypergeometric2F1[I, 1, 1 + I, -(1/E)] - >> E^(2*I)*Hypergeometric2F1[I, 1, 1 + I, -E]) + >> I*E^I*(Hypergeometric2F1[I, 1, 1 + I, -E]/E^(3*Pi) - >> Hypergeometric2F1[I, 1, 1 + I, E^(1 + I)]/E) + >> I*E^(-1 - I)*(-Hypergeometric2F1[I, 1, 1 + I, E^(-1 + I)] + >> E^(2*I)*Hypergeometric2F1[I, 1, 1 + I, E^(1 + I)]) >> >> which, numerically, is >> >> In[359]:= N[%] >> >> Out[359]= -2.7755575615628914*^-17 + 2.7755575615628914*^-17*I >> >> i.e. zero. >> >> On simpler functions like 1, s and s^2 (instead of Exp[I s]) it works out fine, but not so with e.g. Sin[s] in which case we get 0 again (instead of Sinh[2 \[Pi]]). >> >> The integration topic seems to be full of pitfalls in Mathematica... >> >> Best regards, >> Wolfgang >> --- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2838 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

**References**:**Re: Complex path integral wrong***From:*"Kevin J. McCann" <kjm@KevinMcCann.com>