closed form for the following integral which is similar to Laplace transform

Question

I want to find a closed form for this integral:

$\int\limits_{x=0}^{\infty} \exp(-\frac{1}{x})x^n\exp(-sx)dx$

I know that it has closed form for $n=0$ but what about $n\neq0$?

Does anyone have any suggestions or can advise? Thanks

Answer 1

We have: $$ I(s)=\int_{0}^{+\infty}\exp\left(-sx-\frac{1}{x}\right)\,dx = \frac{2}{\sqrt{s}}\,K_1(2\sqrt{s})\tag{1}$$ and your integral can be computed by differentiating both sides of $(1)$ $n$ times with respect to $s$.

Exploiting the Bessel differential equation:

$$ \int_{0}^{+\infty} x^n \exp\left(-sx-\frac{1}{x}\right)\,dx = 2 s^{-\frac{n+1}{2}}\,K_{n+1}(2\sqrt{s}).\tag{2}$$