the Mellin transform
/ MEL-in /
The Fourier transform is the natural tool when a problem is unchanged by shifting (translation invariance); it sorts functions by how they behave under sliding. But some problems on the half-line are unchanged by stretching instead — by rescaling the radius, zooming in or out around the origin. For these, the Mellin transform is the right instrument: it is to scaling what the Fourier transform is to shifting. It shines for problems with corners, wedges, and power-law behaviour near a point.
The Mellin transform of a function f(x) defined for x > 0 is F(s) = integral from 0 to infinity of f(x) x^(s-1) dx, where s is complex. The factor x^(s-1) is the key: under a change of variable x -> a x (a rescaling), it just produces a power of a, which is why the transform turns stretching into multiplication. It is in fact a close relative of the Laplace and Fourier transforms — substituting x = e^(-t) converts a Mellin transform into a two-sided Laplace transform — and its inversion is again a vertical contour integral in the complex s-plane, of the same Bromwich type. Its decisive property for PDEs is that it sends the radial scaling operator x d/dx (and hence the radial Laplacian near a corner, which is built from it) into simple multiplication by s.
Where it earns its place is corner and wedge problems: Laplace's equation in a sector, the stress near a crack tip, diffusion in a wedge — geometries where the solution behaves like a power of the distance r from the corner, r^lambda. The Mellin transform converts the angular PDE into an algebraic problem for the allowed exponents lambda (which appear as the poles of F(s)), and those exponents are exactly the strengths of the singularities at the corner. It is also a workhorse in asymptotics and in number theory (it underlies the analytic continuation of the Riemann zeta function), but in the PDE toolkit it is the specialist tool for scale-invariant, corner-type problems, and is usually met in passing rather than as a daily driver.
Solve Laplace's equation in a wedge of opening angle alpha with the sides held at fixed values. Mellin-transforming the radial variable turns the problem into one for the exponents lambda in solutions of the form r^lambda f(theta); the allowed lambda emerge as poles, and the smallest of them controls how fast the solution (and its gradient) blows up or vanishes at the sharp corner.
Mellin is to scaling what Fourier is to shifting — the tool for corners and power-law behaviour.
Like the Laplace transform, the Mellin integral converges only in a vertical strip of the complex s-plane (set by the function's growth at 0 and at infinity); the inversion contour must lie inside that strip, and the strip can be empty if the two ends of f are incompatible.