       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:= \$Version
> Out= "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= 0
>
> It should have the value
>
> In:= (2 \[Pi] I) Residue[Exp[I s]/(Exp[s] - 1 ), {s, 2 \[Pi] I}]
>
> Out= (2 \[Pi] I) E^(-2 \[Pi])
>
> Without applying FullSimplify the result of the integration is
>
> In:= R = 3*Pi;
> Integrate[
>   Exp[I*s]/(Exp[s] - 1), {s, 1 + I, 1 + I*R, -1 + I*R, -1 + I, 1 + I}]
>
> Out=
> 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:= N[%]
>
> Out= -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
>

```

• Prev by Date: Re: Russian Peasant Multiplication / was question on
• Next by Date: Russian Peasant Multiplication / was question on how
• Previous by thread: Re: Complex path integral wrong
• Next by thread: Re: Complex path integral wrong