Mellin transform

Summary

The Mellin transform sends a function $f$ on $(0, \infty)$ to the function $\mathcal{M}f(s) = \int_0^\infty x^{s-1} f(x) \, dx$ of a complex variable $s$. It is the integral transform adapted to the multiplicative group $(0, \infty)$, in the way the Fourier transform is adapted to the additive group $\R$. It usually converges on a vertical strip of the complex plane, and it turns dilations $x \mapsto ax$ into multiplication by $a^{-s}$. It links power series and Dirichlet series with the gamma function and the Riemann zeta function, and it is a main tool for asymptotic expansions.