Re: Mathematica can't do this double integral

• To: mathgroup at smc.vnet.net
• Subject: [mg48930] Re: Mathematica can't do this double integral
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Thu, 24 Jun 2004 05:35:47 -0400 (EDT)
• Organization: The University of Western Australia
• References: <cb0ufp\$r4a\$1@smc.vnet.net> <cbbb4h\$96k\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <cbbb4h\$96k\$1 at smc.vnet.net>,
Paul Abbott <paul at physics.uwa.edu.au> wrote:

> In article <cb0ufp\$r4a\$1 at smc.vnet.net>,
>
> > It looks like this:
> >
> > Int[Int[E^(a Cos[x]+ b Cos[y]+ k Cos[x - y]) {y, -Pi, Pi}],{x, -Pi, Pi}]
> >
> > Any suggestions anyone?
>
> ... snip
>
> In this way, I have managed to show that the double integral has the
> following nice symmetrical expression:
>
>  (2Pi)^2 (BesselI[0, a] BesselI[0, b] BesselI[0, k] +
>   2 Sum[BesselI[n, a] BesselI[n, b] BesselI[n, k], {n, 1, Infinity}])

Note that since BesselI[-n, a]==BesselI[n, a], this can be written more
elegantly as

(2Pi)^2 Sum[BesselI[n, a] BesselI[n, b] BesselI[n, k],
{n, -Infinity, Infinity}])

and it is actually very easy to obtain this result. Using the generating
function Abramowitz and Stegun 9.6.33 or 9.6.34 (the Fourier series
expansion of Exp[a Cos[t]]) at

http://jove.prohosting.com/~skripty/page_376.htm

one sees that

Exp[a Cos[t]] == Sum[BesselI[n, a] Exp[I n t], {n, -Infinity, Infinity}]

Expanding the three exponential terms in the integrand using this
identity, and then using the orthogonality integral

Integrate[Exp[I m t] Exp[I n t],{t,-Pi,Pi}] == 2 Pi KroneckerDelta[n,-m]

for the integrals over x and y, the result is immediate.

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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