How to evaluate $$\int_{|X|<1}e^{1\over|X|^2-1} dX$$ here $X\in \mathbb R^n$.
I tried like this:
$I=$$\int_{|X|<1}e^{1\over|X|^2-1} dX$ $=\int_0^1e^{1\over r^2-1} n \alpha (n)r^{(n-1)}dr$
$(|X|=r, \alpha(n)=$volume of the n-dimensional unit sphere, $n\alpha(n)r^{(n-1)}=$ surface area of n- dimensional sphere with radius $r$
(I am trying to convert this integration into the form $\int_0^\infty x^{n-1}e^{-x }dx $ so that we may have $\Gamma(n)=\int_0^\infty x^{n-1}e^{-x }dx $)
Let ${1\over1-r^2}=x$, then
$I =n\alpha (n)\int_1^\infty e^{-x} [\frac{x-1}{x}]^{(n-1)/2} (x^2/2)(x/(x-1))^{(1/2)}dx$
Let $x-1=y$, then $I=n\alpha (n)\int_0^\infty e^{-(y+1)} [{y\over y+1}]^{n-1\over2}{(y+1)^2 \over 2}[{y+1\over y}]^{1\over2} dy= n\alpha(n)\int_0^\infty e^{-(y+1)} (y+1)^{{-n\over2}+3}y^{{n\over2}-1}dy$
After this what should I do?