I need help with the calculation of the following integral

$$ \int_{\mathcal{S}}(1-x_1^2)^rdx_1\ldots dx_n $$ where $r>0$ and $$ \mathcal{S} = \left\{(x_1,\ldots,x_n):a-\epsilon\leq x_1+\ldots+x_n\leq a,\;|x_i|\leq1 \;\;\forall i=1,\ldots,n\right\} $$ for $\epsilon>0$.

This question apparently related to the answer of Fabian in Calculating $\int_{\mathcal{S}}x_1^r \, \mathrm dx_1\ldots \, \mathrm dx_n$, however, in this case there are some non analytically integrals in the way...

Thank you

  • $\begingroup$ Can you expand $(1-x_1^2)^r$ using the (generalized) binomial theorem and apply the result from the other post? $\endgroup$ – Eckhard Dec 23 '12 at 14:47
  • $\begingroup$ You could implement the constraints on $x_i$ by hand, i.e., let the integral run from $-1$ to $1$ and use the method in the answer to the other question for the global constraint. $\endgroup$ – Fabian Dec 23 '12 at 16:30
  • $\begingroup$ @Fabian I'm not sure that i'm exactly understand what you mean. Do you mean that I can use the same way just replacing the domain of integration from 0 to infinity to $-1$ to $1$? If so, this is what I tried to do, but, unfortunately it does not work (if i'm not wrong). $\endgroup$ – user39097 Dec 23 '12 at 16:45
  • $\begingroup$ @Eckhard I am not sure that it will help. The problem is with the domain of integration.... $\endgroup$ – user39097 Dec 23 '12 at 16:48
  • 1
    $\begingroup$ Looks like a difficult integral. Maybe you can expand $(1-x^2)^r$ in a binomial series. I am not sure however if the integral over $s$ can be solve analytically in the end. But anyway better to have a single integral to do numerically than $n$ :-) $\endgroup$ – Fabian Dec 23 '12 at 23:01

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