Transformation of Mellin

In Mathematical, the transformation of Mellin is a integral Transformation which can be regarded as the multiplicative version of the Transformation of bilateral Laplace. This integral transformation is strongly connected to the theory of the series of Dirichlet, and is often used into Théorie of the numbers and in the theory of the asymptotic prolongations; it is also strongly connected to the Transformation of Laplace, with the Transformation of Fourier, the theory of the Fonction gamma and with the special functions.

The transformation of Mellin of a function F is

$\backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{M\}\; F\; \backslash \; right\; \backslash \}\; (S)\; =\; \backslash \; varphi\; (S)\; =\; \backslash \; int\_0^\; \{\backslash \; infty\}\; x^s\; F\; (X)\; \backslash \; frac\; \{dx\}\; \{X\}.$

The reverse transformation is

$\backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{M\}\; ^\; \{-\; 1\}\; \backslash \; varphi\; \backslash \; right\; \backslash \}\; (X)\; =\; F\; (X)\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\; I\}\; \backslash \; int\_\; \{Ci\; \backslash \; infty\}\; ^\; \{c+i\; \backslash \; infty\}\; x^\; \{-\; S\}\; \backslash \; varphi\; (S)\; ds.$

The notation supposes that it is a curvilinear Intégrale applying to a vertical line in the complex plan. The conditions under which this inversion is valid are given in the Théorème of inversion of Mellin.

The transformation was named thus into the honor of the Mathématicien Finnish Hjalmar Mellin (1854 - 1933).

Relationships to the other transformations

The Transformation of bilateral Laplace can be defined in terms of transformation of Mellin by

$\backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{B\}\; F\; \backslash \; right\; \backslash \}\; (S)\; =\; \backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{M\}\; F\; (-\; \backslash \; ln\; X)\; \backslash \; right\; \backslash \}\; (S)$

and conversely, we can obtain the transformation of Mellin starting from the transformation of bilateral Laplace by

$\backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{M\}\; F\; \backslash \; right\; \backslash \}\; (S)\; =\; \backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{B\}\; F\; (e^\; \{-\; X\})\; \backslash \; right\; \backslash \}\; (S)$

The transformation of Mellin can be seen as an integration using a core X ^S which respects the multiplicative Mesure of Haar, $\backslash \; frac\; \{dx\}\; \{X\}$, which is invariant under dilation $x\; \backslash \; mapsto\; ax$, i.e. $\backslash \; frac\; \{D\; (ax)\}\{ax\}\; =\; \backslash \; frac\; \{dx\}\; \{X\}$; the transformation of just bilateral Laplace by respecting the additive measurement of Haar $dx\; \backslash ,$, which is an invariant of translation, i.e. $d\; (x+a)\; =\; dx\; \backslash ,$.

We can also define the Transformation of Fourier into terms of transformation of Mellin and vice versa; if we define the transformation of Fourier like above, then

$\backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{F\}\; F\; \backslash \; right\; \backslash \}\; (S)\; =\; \backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{B\}\; F\; \backslash \; right\; \backslash \}\; (is)$

\ left \ {\ mathcal {M} F (- \ ln X) \ right \} (is)

We can also reverse the process and obtain

$\backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{M\}\; F\; \backslash \; right\; \backslash \}\; (S)\; =\; \backslash \; left\; \backslash \; \{\backslash \; mathcal\; \{B\}$

F (e^ {- X}) \ right \} (S) = \ left \ {\ mathcal {F} F (e^ {- X}) \ right \} (- is)

The transformation of Mellin is also connected to the series of Newton or the binomial transformations with the generating Fonction of Poisson, within the meaning of the Cycle Poisson-Mellin-Newton.

Integral of Cahen-Mellin

For $c>0$, $\backslash \; Re\; (there)\; >0$ and $y^\; \{-\; S\}$ on the connects principal, one has

$e^\; \{there\}\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\; I\}$

\ int_ {Ci \ infty} ^ {c+i \ infty} \ Gamma (S) y^ {- S} \; ds

where $\backslash \; Gamma\; (S)$ is the Fonction Gamma of Euler. This integral is known under the name of integral of Cahen-Mellin.

Examples

• the Formule of Perron describes the opposite transformation of Mellin applied to the series of Dirichlet.
• the transformation of Mellin is used in certain evidence of the Fonction of account of the prime numbers and appears in the discussions of the Fonction zeta of Riemann.
• the opposite transformation of Mellin appears commonly in the Rapport of Riesz.

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