Mellin transform
/ MEL-in /
The Fourier transform is built around shifting: it answers what happens when you slide a function along, and it turns translations into phases. But some problems are about scaling, not sliding — about zooming in and out, about behavior near zero versus near infinity. The Mellin transform is the transform tuned to scaling the way the Fourier transform is tuned to shifting; it turns the operation of stretching a function into multiplication, and it is the natural language for power-law behavior.
The Mellin transform of f(x) is F(s) = integral from 0 to infinity of f(x) x^{s minus 1} dx, where s is complex. The kernel x^{s minus 1} is a power of x, which is exactly the eigenfunction of scaling: replacing x by a constant times x just pulls out a power, so a rescaling of the input becomes a simple factor on the output. There is a close family tie: substituting x = e^{minus t} converts the Mellin transform into a two-sided Laplace transform, and a further turn relates it to the Fourier transform — the three are the same machine viewed in additive (Fourier, Laplace) versus multiplicative (Mellin) coordinates. The transform exists on a vertical strip of the complex s-plane, the analog of a region of convergence, fixed by how f behaves at 0 and at infinity.
Its strengths are exactly where scaling rules. It is the transform behind the gamma function (the Mellin transform of e^{minus x} is precisely Gamma(s)) and the Riemann zeta function, the standard tool for extracting the asymptotic expansion of an integral as a parameter goes to 0 or infinity (Mellin-Barnes integrals and the residues of F(s) read off the asymptotic terms), and the right transform for solving differential equations with the scale-invariant Euler-Cauchy structure. In applied work it appears in the analysis of algorithms, in problems with power-law or self-similar behavior, and in summing certain divergent-looking series by analytic continuation in s.
The Mellin transform of e^{minus x} is integral from 0 to infinity of e^{minus x} x^{s minus 1} dx = Gamma(s) — this integral is the definition of the gamma function, so the gamma function literally is a Mellin transform.
The gamma function is the Mellin transform of the decaying exponential — the cleanest illustration of the scale-transform kernel x^{s-1}.
The Mellin transform lives only on the vertical strip where its integral converges, set by the function's growth at 0 and at infinity; quoting a Mellin transform without that strip is as incomplete as a Laplace transform without its region of convergence.