In mathematics, the Mellin transform is an integral transform that may be regarded as the 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 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 function f is
The inverse transform is
The notation implies this is a line integral taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the Mellin inversion theorem.
The transform is named after the Finnish mathematician Hjalmar Mellin.
Read more about Mellin Transform: Relationship To Other Transforms, As A Unitary Operator On L2, In Probability Theory, Applications, Examples
Famous quotes containing the word transform:
“But I must needs take my petulance, contrasting it with my accustomed morning hopefulness, as a sign of the ageing of appetite, of a decay in the very capacity of enjoyment. We need some imaginative stimulus, some not impossible ideal which may shape vague hope, and transform it into effective desire, to carry us year after year, without disgust, through the routine- work which is so large a part of life.”
—Walter Pater (1839–1894)