Multidimensional Mellin transform

Summary

The multidimensional Mellin transform takes a function on the open orthant $(0, \infty)^n$ to a function of $n$ complex variables, by integrating against $x_1^{s_1 - 1} \cdots x_n^{s_n - 1}$. It is the Mellin transform applied in each variable in turn. It turns dilations in each variable into multiplication by powers, and it is used for the asymptotics of multiple integrals and sums.