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MathGroup Archive 2004

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Re: Mathematica can't do this double integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48877] Re: Mathematica can't do this double integral
  • From: astanoff_otez_ceci at yahoo.fr (astanoff)
  • Date: Tue, 22 Jun 2004 05:31:11 -0400 (EDT)
  • References: <cb0ufp$r4a$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Enrique Aguado 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?
> Many many thanks in advance
> Enrique
--

Enrique,
[in case you don't exclude an approximate solution]
If a b and k are "small" (abs typically < 1) a series expansion of exp at 
order 3 at least doesn't seem to give bad outputs compared with NIntegrate 
:

In[1]:=NIntegrate[
  Exp[(a Cos[x]+b Cos[y]+
              k Cos[x-y])] /. a -> .5 /. b -> .5 /. k -> .5, {y,-Pi, 
Pi},{x,-Pi,Pi}]
Out[1]=48.8414

In[2]:=i=Integrate[
      Normal[Series[
            Exp[v],{v,0,3}]] /. v -> (a Cos[x]+b Cos[y]+k Cos[x-y]), 
{y,-Pi,
        Pi},{x,-Pi,Pi}];

In[3]:=i//InputForm
Out[3]//InputForm=(4 + a^2 + b^2 + a*b*k + k^2)*Pi^2

In[4]:=i /. a -> .5 /. b -> .5 /. k -> .5 
Out[4]=48.1143

hth
--
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