Mellin convergence

Summary

A function $f$ on $(0, \infty)$ is Mellin convergent at a complex number $s$ when $x^{s-1} f(x)$ is integrable on $(0, \infty)$. This is exactly the condition under which the Mellin transform $\mathcal{M}f(s)$ is defined as an absolutely convergent integral. It depends only on $\operatorname{Re} s$, and the set of real parts where it holds is an interval. Its interior gives the vertical strip on which the Mellin transform is holomorphic.