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, mathematical statistics, and the theory of asymptotic expansions; it is essentially the same as the Laplace transform and closely related to 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 R + × = ( 0 , ∞ ) {\displaystyle \mathbf {R} _{+}^{\times }=(0,\infty )} is the function M f {\displaystyle {\mathcal {M}}f} of complex variable s {\displaystyle s} given (where it exists, see § Fundamental strip below) by
M { f } ( s ) = φ ( s ) = ∫ 0 ∞ x s − 1 f ( x ) d x = ∫ R + × f ( x ) x s d x x . {\displaystyle {\mathcal {M}}\left\{f\right\}(s)=\varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)\,dx=\int _{\mathbf {R} _{+}^{\times }}f(x)x^{s}{\frac {dx}{x}}.}
Notice that d x / x {\displaystyle dx/x} is a Haar measure on the multiplicative group R + × {\displaystyle \mathbf {R} _{+}^{\times }} and x ↦ x s {\displaystyle x\mapsto x^{s}} is a (in general non-unitary) multiplicative character. The inverse transform is
M − 1 { φ } ( x ) = f ( x ) = 1 2 π i ∫ c − i ∞ c + i ∞ x − s φ ( s ) d s . {\displaystyle {\mathcal {M}}^{-1}\left\{\varphi \right\}(x)=f(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }x^{-s}\varphi (s)\,ds.}
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. Conditions under which this inversion is valid are given in the Mellin inversion theorem. The transform was introduced in 1859 by Bernhard Riemann. The transform is named after the Finnish mathematician Hjalmar Mellin, who developed the first rigorous account in paper published 1897 in Acta Societatis Scientiarum Fennicae.
Relationship to other transforms The two-sided Laplace transform may be defined in terms of the Mellin transform by
B { f } ( s ) = M { f ( − ln x ) } ( s ) {\displaystyle {\mathcal {B}}\left\{f\right\}(s)={\mathcal {M}}\left\{f(-\ln x)\right\}(s)}
and conversely we can get the Mellin transform from the two-sided Laplace transform by
M { f } ( s ) = B { f ( e − x ) } ( s ) . {\displaystyle {\mathcal {M}}\left\{f\right\}(s)={\mathcal {B}}\left\{f(e^{-x})\right\}(s).}
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