Hankel

Hankel refers to several mathematical concepts and objects named after the German mathematician Hermann Hankel (1839–1873). It is also a surname used by various individuals in science and other fields. In mathematics, Hankel functions, Hankel transforms, Hankel matrices, and Hankel operators carry the name and appear in analysis, physics, and signal processing.

Hankel functions are a pair of special functions, denoted usually as H_n^(1)(z) and H_n^(2)(z), that are linear combinations of Bessel functions of the first and second kinds. They solve Bessel’s differential equation and are particularly useful in problems involving outgoing or incoming cylindrical waves, such as diffraction and scattering.

The Hankel transform is an integral transform that generalizes the Fourier transform for radially symmetric functions in two or more dimensions. For a function f(r), the transform involves integrals with Bessel functions J_n, and there is typically a corresponding inverse transform. It is widely used in solving partial differential equations in cylindrical coordinates and in signal processing for radially symmetric data.

A Hankel matrix is a finite or infinite matrix with constant values along each anti-diagonal, meaning a_{i,j} depends only on i+j. This structure makes Hankel matrices useful in system identification, time-series analysis, and numerical linear algebra, where they arise in problems such as moment matching and spectral estimation.

A Hankel operator is a linear operator defined on spaces of analytic or Hardy functions, often described as the compression of multiplication by a fixed function onto a subspace. Hankel operators play a role in complex analysis, operator theory, and control theory.

Historically, the naming honors Hermann Hankel, whose work in analysis and special functions influenced these concepts that remain foundational in applied mathematics.