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Re: Complex path integral wrong

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  • Subject: [mg131413] Re: Complex path integral wrong
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Wed, 3 Jul 2013 22:00:01 -0400 (EDT)
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  • References: <kqomj6$p76$1@smc.vnet.net> <20130701095219.0C6426ABC@smc.vnet.net> <20130703085926.62AE76A91@smc.vnet.net>

The example Hypergeometric2F1[1/2, 1/2, 1, z] reduces to (2/Pi) 
EllipticK[z]. And plotting the imaginary part alone will reveal the 
branch cut. (It's a bit harder to see what's happening on a plot of the 
real part.)

In any case, I thought the original question arose from integrating 
around a contour that missed the real axis entirely. And that the error 
was seen already in integrating along the line segment from 1 + I to 1 + 
3Pi I, which is parallel to the real axis. So I don't see why that 
branch cut should affect the integral there.

On Jul 3, 2013, at 4:59 AM, Dr. Wolfgang Hintze <weh at snafu.de> wrote:

> Am Dienstag, 2. Juli 2013 06:30:05 UTC+2 schrieb Murray Eisenberg:
>> 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
>
> A plot reveals that Mathematica's Hypergeometric2F1 in general has is a cut on the real axis from +1 to +oo as it should be:
>
> f = Hypergeometric2F1[1/2, 1/2, 1, z];
> Plot3D[Re[f /. {z -> x + I y}], {x, 0, 2}, {y, -1, 1}]
> Plot3D[Im[f /. {z -> x + I y}], {x, 0, 2}, {y, -1, 1}]
>
> 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








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