Mellin transformation

In analysis , a branch of mathematics , the Mellin transformation is an integral transformation related to the Fourier transformation . It is named after the Finnish mathematician Hjalmar Mellin .

history

In contrast to the Fourier and Laplace transforms , which were developed to solve physical problems, the Mellin transform was developed in a mathematical context. A first appearance of this integral transformation can be found in a publication by Bernhard Riemann , who used it to investigate his zeta function . A first systematic formulation and investigation of the Mellin transformation and its inverse transformation goes back to the Finnish mathematician R. Hjalmar Mellin. In the area of special functions he developed methods to solve hypergeometric differential equations andderive asymptotic developments .

definition

The Mellin transform of a function defined on the positive real axis is defined as the function ${\displaystyle f\colon \mathbb {R} _{+}\to \mathbb {R} }$

${\displaystyle M_{f}(s):=\int \limits _{0}^{\infty }f(t)t^{s-1}\mathrm {d} t}$

for complex numbers , provided this integral converges. In the literature, the transform can also be found with a normalization factor , i.e. ${\displaystyle s}$ ${\displaystyle {\tfrac {1}{\Gamma (s)}}}$

${\displaystyle {\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }f(t)t^{s-1}\mathrm {d} t.}$

Here is the gamma function . ${\displaystyle \Gamma }$

Inverse transformation

The inverse transformation is under the following conditions

${\displaystyle f(x)={\frac {1}{2\pi \mathrm {i} }}\int \limits _{c-\mathrm {i} \infty }^{c+\mathrm {i} \infty }M_{f}(s)x^{-s}\mathrm {d} s}$

from to for every real with possible. Here and are two positive real numbers. ${\displaystyle M_{f}(s)}$ ${\displaystyle f(x)}$ ${\displaystyle c}$ ${\displaystyle b>c>a>0}$ ${\displaystyle a}$ ${\displaystyle b}$

• the integral is absolutely convergent in the strip ${\displaystyle M_{f}(s)=\int _{0}^{\infty }f(x)x^{s-1}\mathrm {d} x}$ ${\displaystyle S=\{s\in \mathbb {C} \ |\ a<\Re (s)<b\}}$
• ${\displaystyle M_{f}(s)}$is analytical in the strip ${\displaystyle S=\{s\in \mathbb {C} \ |\ a<\Re (s)<b\}}$
• the expression tends towards 0 for and for any value between and evenly ${\displaystyle M_{f}(c\pm \mathrm {i} t)}$ ${\displaystyle t\to \infty }$ ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle b}$
• the function is piecewise continuous on the positive real axis , whereby in the case of discontinuous jumps the mean value of the two-sided limit values ​​should be taken ( step function ) ${\displaystyle f(x)}$

Relationship to the Fourier transform

The Mellin transform is closely related to the Fourier transform . If one substitutes namely in the above integral , one sets and one denotes the inverse Fourier transform of the function with , then for real ${\displaystyle t=e^{x}}$ ${\displaystyle F(x)=f(e^{x})}$ ${\displaystyle F}$ ${\displaystyle {\widehat {F}}}$ ${\displaystyle s}$

${\displaystyle M_{f}(\mathrm {i} s)={\sqrt {2\pi }}{\widehat {F}}(s)}$.

Example for the Dirichlet series

Using the Mellin transformation, a Dirichlet series and a power series can be related to one another. Be there ${\displaystyle f}$ ${\displaystyle F}$

${\displaystyle f(s)=\sum _{n=1}^{\infty }a_{n}n^{-s}}$ and ${\displaystyle F(z)=\sum _{n=1}^{\infty }a_{n}z^{n}}$

with the same . Then applies ${\displaystyle a_{n}}$

${\displaystyle f(s)={\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }F(e^{-t})t^{s-1}\mathrm {d} t}$.

For example , if we put all of them here , then we get the Riemann zeta function , and we get ${\displaystyle a_{n}=1}$ ${\displaystyle f}$

${\displaystyle \zeta (s)={\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }{\frac {t^{s-1}}{e^{t}-1}}\mathrm {d} t}$.

literature

• M. Koecher, A. Krieg, Elliptical functions and modular forms , Springer-Verlag Berlin Heidelberg New York 1998, ISBN 3-540-63744-3 .
• EC Titchmarsh, Introduction to the Theory of Fourier Integrals , Chelsea Publishing Company, 3rd Edition 1986, ISBN 978-0-8284-0324-5 .
• D. Zagier, Zeta functions and square bodies , Springer-Verlag Berlin Heidelberg New York 1981, ISBN 3-540-10603-0 .

Web links

• Eric W. Weisstein : Mellin Transform . In: MathWorld (English).

Individual evidence

1. ↑ Jacqueline Bertrand, Pierre Bertrand, Jean-Philippe Ovarlez: The Transforms and Applications Handbook . Ed .: Alexander D. Poularikas. 2nd Edition. CRC Press, 2000, ISBN 978-0-8493-8595-7 , chapter 11.1.