Mellin transform

Under the Mellin transform is understood in the analysis, a branch of mathematics, a related to the Fourier transform integral transformation. It is named after the Finnish mathematician Hjalmar Mellin.

History

In contrast to Fourier and Laplace transforms, which were devised to solve physical problems Mellin transformation is designed in a mathematical context. A first occurrence of this integral transformation can be found in a paper by Bernhard Riemann, she began to study its zeta function. A first systematic formulation and examination of the Mellin transform and its inverse transform goes back to the Finnish mathematician R. Hjalmar Mellin. In the area of ​​special functions, he developed methods to solve hypergeometric differential equations and derive asymptotic expansions.

Definition

The Mellin transform of a defined function on the positive real axis is defined as the integral operator

For complex numbers, if this integral converges. In the literature one finds the transform with a scaling factor, ie

Here is the gamma function.

Inverse transformation

Under the following conditions is the inverse transformation

From to for every real with possible. Here are and two positive real numbers.

• The integral is absolutely convergent in the strip
• Is analytic in the strip
• The term sought for and any value between 0 and uniformly to
• The function is on the positive real axis piecewise continuous, and in the case of discontinuous jumps, the average value of the two-sided limit is to be taken ( step function )

Relation to the Fourier transform

The Mellin transform is closely related to the Fourier transform. Substituting namely in the above integral is given to and is defined as the Fourier transform of the function, then

Example of Dirichlet series

By means of the Mellin transform can create relationships between a Dirichlet series and a power series. There are

With the same. Then we have

If, for example, all herein, then the Riemann zeta function, and we obtain