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 function f is

The inverse transform is

The notation implies this is a line integral taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the Mellin inversion theorem.

The transform is named after the Finnish mathematician Hjalmar Mellin.

The two-sided Laplace transform may be defined in terms of the Mellin transform by

and conversely we can get the Mellin transform from the two-sided Laplace transform by

The Mellin transform may be thought of as integrating using a kernel x^s with respect to the multiplicative Haar measure, ${\displaystyle {\frac {dx}{x}}}$, which is invariant under dilation ${\displaystyle x\mapsto ax}$, so that ${\displaystyle {\frac {d(ax)}{ax}}={\frac {dx}{x}};}$ the two-sided Laplace transform integrates with respect to the additive Haar measure ${\displaystyle dx}$, which is translation invariant, so that ${\displaystyle d(x+a)=dx}$.

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