I would like to solve:
$$\int_{\mathbf{R}^n} \text{exp}\left(-\left(\sum_i |x_i|^a\right)^b\right)\;dx_1...dx_n$$
I have no idea how to tackle this integral. For the special case of $a=1$ I obtained this formula by trial and error (I can't prove it):
$$\frac{2^n}{b \cdot (n-1)!} \Gamma\left(\frac{n}{b}\right)$$
For the special case of $b=1$ I can proceed as follows:
$$\int_{\mathbf{R}^n} \text{exp}\left(-\sum_i |x_i|^a\right)\;dx_1...dx_n = 2^n\int_0^\infty...\int_0^\infty \text{exp}\left(-\sum_i x_i^a\right)\;dx_1...dx_n\\=2^n\prod_i{\int_0^\infty \text{exp}\left(-x_i^a\right) dx_i} = \frac{2^n}{a^n}\prod_i{\int_0^\infty \text{exp}\left(-y_i\right)y_i^{\frac{1}{a}-1} dy_i} = \left(\frac{2\Gamma(1/a)}{a}\right)^n,$$
where I used a change of variables: $y_i = x_i^a \to \frac{dx_i}{dy_i} = \frac{1}{a}y_i^{\frac{1}{a}-1}$.
How do I find a closed form for arbitrary $a,b$ ?
Any hints would be much appreciated.
A general formula here, for $a>0$ and "good enough" $f:\mathbb{R}_+\to\mathbb{R}$: $$\int_{\mathbb{R}^n}f\left(\sum_{i=1}^n|x_i|^a\right)\,dx_1\cdots dx_n=\frac{\big[2\Gamma(1+1/a)\big]^n}{\Gamma(1+n/a)}\int_0^\infty f(x^{a/n})\,dx.$$ Applied to $f(x)=e^{-x^b}$, this easily gives $$\int_{\mathbb{R}^n}\exp\left[-\Big(\sum_{i=1}^n|x_i|^a\Big)^b\right]\,dx_1\cdots dx_n=\big[2\Gamma(1+1/a)\big]^n\frac{\Gamma(1+n/ab)}{\Gamma(1+n/a)}.$$