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
Relationship to other transforms
The two-sided Laplace transform may be defined in terms of the Mellin transform by
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 xs 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 , which is translation invariant, so that .
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
We may also reverse the process and obtain
- Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
- 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. ISBN 0-8493-2876-4