Editing Mellin transform (section)

{{Short description|Mathematical operation}}
In [[mathematics]], the '''Mellin transform''' is an [[integral transform]] that may be regarded as the [[multiplicative group|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 expansion]]s; it is closely related to the [[Laplace transform]] and the [[Fourier transform]], and the theory of the [[gamma function]] and allied [[special function]]s.

The Mellin transform of a complex-valued function {{mvar|f}} defined on <math>\mathbf R^{\times}_+= (0,\infty) </math> is the function <math>\mathcal M f</math> of complex variable <math>s</math> given (where it exists, see [[#Fundamental strip|Fundamental strip]] below) by
<math display="block">\mathcal{M}\left\{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}.</math>
Notice that <math>dx/x</math> is a [[Haar measure]] on the multiplicative group <math>\mathbf R^{\times}_+</math> and <math>x\mapsto x^s</math> is a (in general non-unitary) [[multiplicative character]].
The inverse transform is 
<math display="block">\mathcal{M}^{-1}\left\{\varphi\right\}(x) = f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s)\, ds.</math>
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 [[Finland|Finnish]] mathematician [[Hjalmar Mellin]], who introduced it in a paper published 1897 in ''Acta Societatis Scientiarum Fennicae.''<ref>{{Cite journal|last=Mellin|first=Hj.|title=Zur Theorie zweier allgemeinen Klassen bestimmter Integrale|journal=Acta Societatis Scientiarum Fennicae|volume=XXII |issue=2|pages=1–75}}</ref>