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Topic: Complex path integral wrong
Replies: 7   Last Post: Jul 3, 2013 4:50 AM

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Murray Eisenberg

Posts: 2,105
Registered: 12/6/04
Re: Complex path integral wrong
Posted: Jul 2, 2013 12:30 AM
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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 (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@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









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