Re: Complex path integral wrong

*To*: mathgroup at smc.vnet.net*Subject*: [mg131369] Re: Complex path integral wrong*From*: "Kevin J. McCann" <kjm at KevinMcCann.com>*Date*: Mon, 1 Jul 2013 05:52:19 -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>

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 >

**Follow-Ups**:**Re: Complex path integral wrong***From:*"Dr. Wolfgang Hintze" <weh@snafu.de>

**Re: Complex path integral wrong***From:*Murray Eisenberg <murray@math.umass.edu>