Inverse Mellin transform

Summary

The inverse Mellin transform turns a function $\varphi$ on a vertical line $\operatorname{Re} s = c$ into the function $x \mapsto \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} \varphi(s) \, ds$ on $(0, \infty)$. By the Mellin inversion theorem, it undoes the Mellin transform under suitable hypotheses. In logarithmic variables it is an inverse Fourier transform, and when $\varphi$ is meromorphic it can often be computed with the residue theorem.