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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>,
> Enrique Aguado <l.e.aguado at leeds.ac.uk> wrote:
>
> > 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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