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?