
Re: Complex path integral wrong
Posted:
Jul 2, 2013 12:30 AM


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@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 (64bit) (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@math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 5491020 (H) University of Massachusetts 413 5452838 (W) 710 North Pleasant Street fax 413 5451801 Amherst, MA 010039305

