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MathGroup Archive 2000

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Re: Difficult integral: correction

  • To: mathgroup at smc.vnet.net
  • Subject: [mg23454] Re: Difficult integral: correction
  • From: "Rasmus Debitsch" <debitsch at Zeiss.de>
  • Date: Thu, 11 May 2000 00:54:06 -0400 (EDT)
  • Organization: Customer of UUNET Deutschland GmbH
  • References: <8f559t$49c@smc.vnet.net> <8fb0th$hid@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

with

f =Expand[TrigToExp[Sin[x]^(2 rho)] Exp[I n x]]
Integrate [f,{x,0,2 Pi}]

i had no problem to compute the integral for rho=1,....,12 in a few minutes
(Mathematica 4, WinNT, PIII 500, 256MB).

Perhaps the TrigToExp is the trick.

A other guess for the integral is :

Let i[rho] = Integrate [(Sin[x])^(2 rho) Exp[I n x],{x,0,2 Pi}]

and make the observation

i[rho+1] = -2 i[rho] (rho+1) / (2rho+1)/(n^2-(2rho+2)^2)

which will be a fast way to compute the integral for higher values of rho.
Perhaps it is possible to prove this - I haven't tried it.

Regards

--

        Rasmus Debitsch
        Carl Zeiss Lithos GmbH
        Carl Zeiss Strasse
        D-73447 Oberkochen

        eMail : debitsch at zeiss.de

Marco de Innocentis <mdeinnocentis at hotmail.com> schrieb in im Newsbeitrag:
8fb0th$hid at smc.vnet.net...
> I am sorry, I made a silly mistake: rho was supposed to be part
> of the exponential, so
>
> > Integrate [(Sin[x])^[2 rho] Exp[I n x],{x,0,2 Pi}]
>
> Thanks,
>
> Marco
>
> In article <8f559t$49c at smc.vnet.net>,
>   Marco de Innocentis <mdi11 at hotmail.com> wrote:
>
>
> > I need to calculate the following integral
> >
> > Integrate [(Sin[x])^2 rho Exp[I n x],{x,0,2 Pi}]
> >
> > as a function of the two parameters rho and n. I have tried it
> > for rho = 1, 4, 8, 10, and it works, but it doesn't work at all
> > for rho = 12. I left the computer on all night but only got the
> > same expression as a result.
> > Any suggestions?
> > Thanks
> >
> > Marco
> >
> > Sent via Deja.com http://www.deja.com/
> > Before you buy.
> >
> >
>
> --
> http://www.thehungersite.com
>
>
> Sent via Deja.com http://www.deja.com/
> Before you buy.
>
>


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end



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