Can anyone help me get this Laplace transform of

Question

$f(t) = a + b \exp(-c \cdot t ^ d) $, where $a,b,c,d$ are constants, and $d$ is power of $t$.

Answer 1

It boils down to computing $J = \int_0^\infty \exp(-st - t^d)\, dt$. I'll assume $d > 1$. Write as a series in $-s$: $J = \sum_{n=0}^\infty \int_0^\infty \frac{(-s)^n t^n}{n!} \exp(-t^d)\, dt = \sum_{n=0}^\infty \frac{\Gamma((n+1)/d)}{n!\, d} (-s)^n$. I don't know if there is a "closed form" for this, but for each positive integer $d$ it can be written in terms of hypergeometric functions.