Mellin transform

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.

The Mellin transform of a complex-valued function f defined on ${\displaystyle \mathbf {R} _{+}^{\times }=(0,\infty )}$is the function ${\displaystyle {\mathcal {M}}f}$ of complex variable ${\displaystyle s}$ given (where it exists, see Fundamental strip below) by $${\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)\,dx=\int _{\mathbf {R} _{+}^{\times }}f(x)x^{s}{\frac {dx}{x}}.}$$ Notice that ${\displaystyle dx/x}$ is a Haar measure on the multiplicative group ${\displaystyle \mathbf {R} _{+}^{\times }}$ and ${\displaystyle x\mapsto x^{s}}$ is a (in general non-unitary) multiplicative character. The inverse transform is $${\displaystyle \left\{{\mathcal {M}}^{-1}\varphi \right\}(x)=f(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }x^{-s}\varphi (s)\,ds.}$$ The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem.

The transform is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper published 1897 in Acta Societatis Scientiarum Fennicæ.^[1]