# Difference between revisions of "Mellin transform"

The integral transform

$$M(p)=\int\limits_0^\infty f(t)t^{p-1}dt,\quad p=\sigma+i\tau.$$

The substitution $t=e^{-z}$ reduces it to the Laplace transform. The Mellin transform is used for solving a specific class of planar problems for harmonic functions in a sectorial domain, of problems in elasticity theory, etc.

The inversion theorem. Suppose that $\tau^{\sigma-1}f(\tau)\in L(0,\infty)$ and that the function $f(\tau)$ has bounded variation in a neighbourhood of the point $\tau=t$. Then

$$\frac{f(t+0)-f(t-0)}{2}=\frac{1}{2\pi i}\lim_{\lambda\to\infty}\int\limits_{\sigma-i\lambda}^{\sigma+i\lambda}M(s)t^{-s}ds.$$

The representation theorem. Suppose that the function $M(\tau+iu)$ is summable with respect to $u$ on $(-\infty,+\infty)$ and has bounded variation in a neighbourhood of the point $u=t$. Then

$$\frac{M(\sigma+i(t+0))+M(\sigma+i(t-0))}{2}=\lim_{\lambda\to\infty}\int\limits_{1/\lambda}^\lambda f(x)x^{\sigma+it-1}dx,$$

where

$$f(x)=\frac{1}{2\pi i}\int\limits_{\sigma-i\infty}^{\sigma+i\infty}M(s)x^{-s}ds.$$

How to Cite This Entry:
Mellin transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mellin_transform&oldid=23896
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article