Mellin transform

The Mellin transform converts a function into a function of a complex exponent by integrating it against powers of its variable. It is especially suited to scale-invariant problems.

Mellin inversion theorem: A theorem that reconstructs a function from its Mellin transform by contour integration. It gives the conditions and integral formula for recovering the original function.
Bilateral Laplace transform: An integral transform integrating a function over the real line against complex exponentials. The substitution from positive variables to logarithmic coordinates connects it to Mellin analysis.
Mellin–Barnes integral: A contour integral representation whose integrand contains products of gamma functions and complex powers. It uses Mellin inversion to represent special functions and evaluate difficult integrals.
Hjalmar Mellin: A Finnish mathematician known for work on special functions and integral transforms. His work established the transform now bearing his name.
Fundamental strip: The vertical strip of complex exponents where a Mellin transform converges. The transform’s convergence region determines where inversion and analytic continuation begin.
Fourier transform: An integral transform that decomposes a function into oscillations indexed by frequency. On logarithmic coordinates, the Mellin transform becomes a Fourier transform.
Analytic number theory: The study of arithmetic questions using methods from mathematical analysis. Mellin transforms relate arithmetic sums to zeta and L-functions.
Niels Henrik Abel: A Norwegian mathematician whose work shaped the theory of equations, series, and integrals. Abelian integral methods form part of the historical setting for Mellin’s analysis.
Mellin convolution: An integral operation on functions of positive variables that becomes multiplication under the Mellin transform. It plays the role that ordinary convolution plays for the Fourier transform.
Integral transform: A mapping that converts a function into another function through integration against a kernel. The Mellin transform is one member of this broad family.

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