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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ... In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ... In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by: The lower limit of 0− is short notation to mean and assures the inclusion of the entire dirac delta function... The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ... In mathematics, several functions are important enough to deserve their own name. ...

The Mellin transform of a function f is

The inverse transform is

The notation implies this is a path integral taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the Mellin inversion theorem. This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics, the Mellin inversion formula tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function. ...

The transform is named after the Finnish mathematician Robert Hjalmar Mellin (1854 - 1933). Robert Hjalmar Mellin (1854-1933) was a Finnish mathematician. ...

Relationship to other transforms

The two-sided Laplace transform may be defined in terms of the Mellin transform by In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. ...

and conversely we can get the Mellin transform from the two-sided Laplace transform by

The Mellin transform may be thought of as integrating using a kernel x^s with respect to the multiplicative Haar measure, , which is invariant under dilation , so that ; the two-sided Laplace transform integrates with respect to the additive Haar measure dx, which is translation invariant, so that d(x + a) = dx. In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...

We also may define the Fourier transform in terms of the Mellin transform and vice-versa; if we define the two-sided Laplace transform as above, then The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

We may also reverse the process and obtain

External links

• Tables of Integral Transforms (http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm) at EqWorld: The World of Mathematical Equations.

References

• Paris, R. B., and Kaminsky, D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001.
• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.