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Concept
Mellin transform
Summary
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, 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.
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 xs with respect to the multiplicative Haar measure,
which is invariant
under dilation , so that
the two-sided Laplace transform integrates with respect to the additive Haar measure , which is translation invariant, so that .
We also may define the Fourier transform in terms of the Mellin transform and vice versa; in terms of the Mellin transform and of the two-sided Laplace transform defined above
We may also reverse the process and obtain
The Mellin transform also connects the Newton series or binomial transform together with the Poisson generating function, by means of the Poisson–Mellin–Newton cycle.
The Mellin transform may also be viewed as the Gelfand transform for the convolution algebra of the locally compact abelian group of positive real numbers with multiplication.
The Mellin transform of the function is
where is the gamma function.
Official source
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