Mellin transform

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 closely related to the Fourier transform and to the theory of the gamma function and similar special functions. The Mellin transform of a complex-valued function f {\displaystyle f} defined on R + × = ( 0 , ∞ ) {\displaystyle \mathbb {R} _{+}^{\times }=(0,\infty )} is the function M f {\displaystyle {\mathcal {M}}f} of a complex variable s {\displaystyle s} defined 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 _{\mathbb {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 \mathbb {R} _{+}^{\times }} and x ↦ x s {\displaystyle x\mapsto x^{s}} is a (generally non-unitary) multiplicative character. The inverse transform is given by 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.} This is a line integral over a vertical line in the complex plane, whose real part c {\displaystyle 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...