Re: Integrate product of Gaussian*Sin

• To: mathgroup at smc.vnet.net
• Subject: [mg6916] Re: [mg6855] Integrate product of Gaussian*Sin
• From: seanross at worldnet.att.net
• Date: Fri, 25 Apr 1997 14:00:44 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```James Perry wrote:
>
> Hi,
>
>         Can anyone suggest a good change of variables to carry out the
> integration
>
>         Integrate[Exp[-alpha*(x-a)^2*Sin[n Pi x/L],{x,-Infinity,Infinity}]
>
>         I can't find this form in my integral tables (I'm going to check
> the library today for a more comprehensive list, I might find a form that I
> can convert my expression to), and Mathematica can't find a solution unless
> I take the limits of the integral {x,-c,c}. However, treated as an improper
> integral
>
>         Integrate[Exp[-alpha*(x-a)^2*Sin[n Pi x/L],{x,-c,c}]
>         Limit[%,c->Infinity]
> or      Limit[%,c->-Infinity]
>
>         Still does not give a solution, since the answer to the integral
> (with limits {x,-c,c}) is a combination of Erf[x] and Erfi[x], and the
> Limit[Erfi[x],x->+/- Infinity]->+/- Infinity. The Erf[x] has a limit of +/-
> 1 as x->+/- Infinity.
>         I'm not sure if there is a solution to this, anyone with
> Gaussian-type function experience?
>
> Thank you
> Jim

Re-write the Sin as a complex exponential.  You will then have two
integrals both of which are quadratic in the exponent, but with a
complex coefficient.  The problem here will not be to get a result, but
to interpret it.  ie. what exactly is the error function for complex
arguments?

```

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