---
tag: 92JY94ZJD8
kind: concept
name: Multidimensional Mellin transform
msc: [44A30, 44A15]
specializes: [4PVP1C3JZG]
sources:
  - url: https://en.wikipedia.org/wiki/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|N5PVGRJE1T]] 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.

## Explanations

### L1 human

The one-variable Mellin transform measures how a function behaves under stretching of its variable.
In several variables each variable may be stretched separately, and one complex exponent is attached to each.
For example, $e^{-x_1 - x_2}$ has the two-variable Mellin transform $\Gamma(s_1)\Gamma(s_2)$, the product of two gamma functions.
The logarithmic change of variables $x_j = e^{-t_j}$ turns it into a multidimensional Laplace or Fourier transform.