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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```

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