Definition:Mellin Transform

Definition
Let $f \left({t}\right): \R_{\ge 0} \to \C$ be a function of a real variable $t$.

The Mellin Transform of $f$, denoted $\mathcal M \left({f}\right)$ or $\phi$, is defined as:
 * $\displaystyle \mathcal M \left\{{f \left({t}\right)} \right\} \left({s}\right) = \phi \left({s}\right) = \int_0^{\to +\infty} t^{s-1} f \left({t}\right) \mathrm d t$

wherever this improper integral exists.

Here $\mathcal M \left({f}\right)$ is a complex function of the variable $s$.