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

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Dr. Wolfgang Hintze

Posts: 195
Registered: 12/8/04
Complex path integral wrong
Posted: Jun 30, 2013 3:26 AM
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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




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