Hankel transform
/ HAHN-kel /
The Fourier transform is the right tool when a problem is laid out along a straight line, because its building blocks are the waves e^{i k x}. But many problems in physics are round — a vibrating circular drumhead, heat spreading from a point in a disk, light through a circular lens. For these, the natural building blocks are not straight sine waves but Bessel functions, the radial standing-wave patterns that respect circular symmetry. The Hankel transform is the Fourier transform's circular relative, built from Bessel functions instead of exponentials.
Precisely, the Hankel transform of order nu sends a function f(r) of a radial variable to F(k) = integral from 0 to infinity of f(r) J_nu(k r) r dr, where J_nu is the Bessel function of the first kind of order nu, and the extra factor of r is the radial part of the area element. Its inverse has the same form, which makes the transform nearly self-reciprocal. It arises naturally: if you take an ordinary two-dimensional Fourier transform of a function that depends only on the distance from the origin (a circularly symmetric function), the angular integral collapses and what survives is exactly the order-zero Hankel transform. So the Hankel transform is the two-dimensional Fourier transform with the rotational symmetry already factored out.
Its job is to do for cylindrical and polar problems what the Fourier transform does for Cartesian ones: turn the radial part of the Laplacian into multiplication and reduce a partial differential equation to an algebraic one. It is the standard route for boundary-value problems on disks and infinite half-spaces with circular symmetry, for diffraction by circular apertures in optics (where the diffraction pattern of a round hole is the Hankel transform of the aperture), and for axisymmetric heat and wave problems. The order nu of the Bessel function is set by the angular part of the problem, so different angular modes use different-order Hankel transforms.
The diffraction pattern of a uniformly lit circular aperture of radius a is its order-zero Hankel transform, which gives the Airy pattern: a bright central disk surrounded by faint rings, with intensity proportional to (J_1(k a)/(k a))^2 — the reason a telescope's image of a star is a tiny disk, not a point.
A round aperture's diffraction pattern is its Hankel transform — the Airy disk that sets a telescope's resolution.
The Hankel transform is the radial fingerprint of the multi-dimensional Fourier transform, not a separate idea: it only applies to the radial part of a circularly or axially symmetric problem, and the correct order nu must match the angular mode or the answer is wrong.