the Hankel transform
/ HAHN-kel /
When a problem has circular or spherical symmetry — heat spreading out from a point on a flat plate, a vibrating circular drumhead, ripples from a stone dropped in a still pond — the solution depends only on the distance r from the centre, not on the angle. Using a full two-dimensional Fourier transform on such a problem wastes effort, because you already know the answer is radial. The Hankel transform is the version of the Fourier transform built for exactly these radially symmetric problems, working directly in the radial variable r.
Where the Fourier transform uses oscillating exponentials, the Hankel transform of order zero uses a Bessel function as its kernel: it is integral from 0 to infinity of f(r) J_0(k r) r dr, where J_0 is the Bessel function of the first kind of order zero and the extra factor of r is the natural area weight in polar coordinates (a thin ring at radius r has area proportional to r). The reason it appears is structural: when you write the two-dimensional Laplacian in polar coordinates and demand no angular dependence, the radial part is exactly Bessel's operator, so its natural 'frequencies' are Bessel functions rather than sines and cosines. Just like the Fourier transform turns u_xx into -xi^2 u-hat, the order-zero Hankel transform turns the radial Laplacian acting on an angle-independent function into multiplication by -k^2 — a derivative becomes a number again.
So the workflow is familiar: a radial PDE, Hankel-transformed, becomes an ODE in the transform variable k, which you solve and then invert (the Hankel transform is conveniently its own inverse). It is the natural tool for diffusion or waves on a disk or the infinite plane with a point source, and for many problems in optics and electromagnetism with circular apertures. The honest caveat: it only helps when the problem really is radially symmetric; with angular dependence you must first expand in angular modes (e^(i m theta)), and each mode m then uses the Hankel transform of order m with kernel J_m instead of J_0.
Drop a unit of heat at the centre of an infinite flat plate and let it diffuse: u_t = k(u_rr + (1/r) u_r). Hankel-transform in r and it becomes u-hat_t = -k k_var^2 u-hat (writing k_var for the transform variable), giving u-hat = e^(-k k_var^2 t). Inverting reproduces the two-dimensional radial Gaussian heat kernel — the same diffusion you would get from the full 2D Fourier transform, but reached more directly.
For radial problems the Bessel kernel replaces the exponential, turning the radial Laplacian into -k^2.
The factor of r in the integral is not optional decoration — it is the polar area element, and dropping it breaks the inversion. The transform also requires genuine radial symmetry (or one angular mode at a time); it is not a general two-dimensional transform.