For a given integer $N$ and some function $f(x,y)$ I want to be able to do an integration of the kind,

$\int_{0} ^{\infty} \prod_ {i=1}^N da_i e^{[\sum_{n=1}^{\infty} \frac{1}{n} \left[ N + f(x^n,y^n)\sum _{i,j=1}^N cos(n(a_i-a_j)) \right ] ] }$

I guess that I have to anyway remove the N/n term in the exponent for anything to converge. A typical $f$ will be like $f(x,y) = -1 + 2\frac{(\sqrt{y} - 1/\sqrt{y})(1/\sqrt{x} - \sqrt{x})}{1/x - x}$.

I expect the answer to be a power-series in $x$ and $y$.

For even N=2 I couldn't manage to get an answer from Mathematica! I tried cutting off the sum over $n$ to some finite number of terms rather than an infinite sum and still I couldn't get an answer.

The code is being pasted below,

Integrate[
Exp[Sum[
((-1 + (2 (x^(-n/2) + x^(n/2)) (y^(-n/2) + y^(n/2)))/(x^(-n) - x^n))
* (2 + 2 Cos[(a1 - a2) n])
)/n,
{n, 1, 2}
]],
{a1, 0, Infinity}, {a2, 0, Infinity}
]


For larger N I don't know how to make it on its own generate that double sum over cosine for a list of integration variables. For small N I can may be manually type the expression but that can't be a sustainable solution.

I would like to know how to do this integral.

-
Please list the MMA code you used to try to get the N=2 case. –  tkott Oct 10 '12 at 0:58
@tkott I did a copy-paste of the expression but I don't know how to make it look readable here. Can you help? –  user6818 Oct 10 '12 at 14:32
Short answer might be: use NIntegrate with values for x and y. Mathematica is can't figure out an analytic expression for the Sum[f[x^n,y^n],{n,1,Infinity}] term, so I don't see how it can get an analytic expression for the full integral. Additionally, while Sum[2 Cos[(a1 - a2) n]/n, {n, 1, \[Infinity]}] has an analytic expression (apparently), Sum[f[x^n,y^n] Cos[(a1 - a2) n]/n, {n, 1, \[Infinity]}] does not appear to have one. You'll need to simplify things yourself before throwing mathematica at it I think. –  tkott Oct 10 '12 at 16:54
@tkott The point is that I need the answer as a power-series in $x$ and $y$. And I have little hope that this sum over $n$ can have any closed form expression - in general $N$ will be very large and $f$ will be a more complicated function - so you are saying that Mathematica is first trying to do the sum over $n$ and then integrate and since it can't do the first it just gives up there itself? –  user6818 Oct 10 '12 at 20:34
yes, essentially. You might be able to play around with Assuming if you know some properties of $x$ and $y$ such as between 0 and 1, or something similar. The general math problem seems a bit hard. Have you tried asking math.stackexchange.com for suggestions on simplifying the expression? –  tkott Oct 10 '12 at 21:50
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