Calculate: $f(a)=\int\limits_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x$

Question

Given the following function:

$$ f(a)=\int_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x $$

For which values of $a$ is it possible to give an exact value for this function? I only know $f(2)=\sqrt{\pi}$, $f(1)=2$, $f(0)=\infty$, $f(\infty)=2$. Are there any other values which can be calculated, and does there exist a formula to calculate any value of this function?

Answer 1

We may write $$\int_{-\infty}^{\infty} e^{-|x|^a}\,dx = 2\int_0^\infty e^{-x^a}\,dx$$

Using the substitution $u = x^a \iff du = ax^{a-1}\,dx \iff \frac1au^{\frac{1}{a}-1}adu=dx$ and assuming $a>0$

$$\begin{align*}2\int_0^\infty e^{-x^a}\,dx&=\frac{2}a{}\int_0^\infty u^{\frac1a-1}e^{-u}\,du \\&= \frac{2}{a}\Gamma\left(\frac{1}{a}\right) \\&= 2\Gamma\left(\frac1a + 1\right)\end{align*}$$

In the case $a\leq0$ the function obviously diverges.