Mellin transform

For a function \(f:\mathbb{R}\rightarrow \mathbb{C}\), the Mellin transform is defined as \[\begin{equation} M\{f\}(s) = \int_{0}^{\infty} x^{s-1} f(x) dx. \end{equation}\]

There are some theorems about when the above integral convergges and when is the Mellin transform \(M\{f\}\) become analytic. See wikipedia.

There’s also the following inversion formula. \[\begin{equation} f(x) = \int_{c-i \infty}^{c+ i \infty}x^{-s} M\{f\}(s) ds, \end{equation}\] for \(c \in \mathbb{R}\) large enough.