How can the integral of exp(-x^n) be proven using polar coordinates?

[nonequilibrium]

Hello,

How do I prove or find a proof for $\int e^{-(x_1^2+\dots+x_\nu^2)^{k/2}} \mathrm d x = \frac{\pi^{\nu/2} \Gamma \left( \frac{\nu}{k}+1 \right)}{\Gamma \left( \frac{\nu}{2} +1\right)}$?

EDIT: I think I have it, and as I can't delete this thread, I will sketch the solution:
first of all, notice that proving it for the case of $\nu = 1$ is enough, because we can always go back to that specific case by using polar coordinates (replacing $x_1^2 + \dots + x_\nu^2 \mapsto r^2$); and then:

$\int_{-\infty}^{+\infty} e^{-|x|^k} \mathrm dx = 2 \int_{0}^{+\infty} e^{-x^k} \mathrm dx$ and if we substitute with y=x^k => $\mathrm d y = k y^{(k-1)/k} \mathrm d x$, this is equal to $2\int_0^{+\infty} e^{-y} y^{\frac{1}{k}-1} \frac{1}{k} \mathrm d y = \frac{2}{k} \Gamma \left(\frac{1}{k} \right)$ and using $x \Gamma(x) = \Gamma(x+1)$ we get:
$\int_{-\infty}^{+\infty} e^{-|x|^k} \mathrm dx = 2 \Gamma \left( \frac{1}{k} + 1 \right) = \frac{\pi^{1/2} \Gamma \left( \frac{1}{k}+1 \right)}{\Gamma \left( \frac{1}{2} +1\right)}$

 

[fresh_42]

Thank's a lot for sharing the solution!