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 complex-valued function f defined on
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.
The transform is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper published 1897 in Acta Societatis Scientiarum Fennicæ.
The Mellin transform may be thought of as integrating using a kernel xs with respect to the multiplicative Haar measure,
the two-sided Laplace transform integrates with respect to the additive Haar measure
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.
However, by defining it to be zero on different sections of the real axis, it is possible to take the Mellin transform.
It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function,
is a generalized Gaussian distribution without the scaling factor.)
by this formal identity involving the Mellin transform:[4]
define the left endpoint of its fundamental strip, and the asymptotics of the function as
The properties in this table may be found in Bracewell (2000) and Erdélyi (1954).
The integration on the right hand side is done along the vertical line
that lies entirely within the overlap of the (suitable transformed) fundamental strips.
In the study of Hilbert spaces, the Mellin transform is often posed in a slightly different way.
(see Lp space) the fundamental strip always includes
In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables.
[8] If X is a random variable, and X+ = max{X,0} denotes its positive part, while X − = max{−X,0} is its negative part, then the Mellin transform of X is defined as[9]
of a random variable X uniquely determines its distribution function FX.
For example, the 2-D Laplace equation in polar coordinates is the PDE in two variables:
with a Mellin transform on radius becomes the simple harmonic oscillator:
Now let's impose for example some simple wedge boundary conditions to the original Laplace equation:
The Mellin transform is widely used in computer science for the analysis of algorithms[12] because of its scale invariance property.
The magnitude of the Mellin Transform of a scaled function is identical to the magnitude of the original function for purely imaginary inputs.
The magnitude of a Fourier transform of a time-shifted function is identical to the magnitude of the Fourier transform of the original function.
In quantum mechanics and especially quantum field theory, Fourier space is enormously useful and used extensively because momentum and position are Fourier transforms of each other (for instance, Feynman diagrams are much more easily computed in momentum space).
In 2011, A. Liam Fitzpatrick, Jared Kaplan, João Penedones, Suvrat Raju, and Balt C. van Rees showed that Mellin space serves an analogous role in the context of the AdS/CFT correspondence.
[13][14][15] Below is a list of interesting examples for the Mellin transform: