Hankelmuunnos

Hankelmuunnos, known in English as the Hankel transform, is an integral transform used primarily in mathematical analysis, applied mathematics, and engineering, especially in contexts involving radially symmetric functions. It is named after the German mathematician Hermann Hankel. The transform is closely related to the Fourier and Laplace transforms but is particularly suited for functions defined over radial domains, such as in problems with cylindrical or spherical symmetry.

The Hankel transform of a function \(f(r)\), where \(r \geq 0\), is defined as an integral involving the Bessel function of the first kind, \(J_\nu(r)\), of order \(\nu\). The general form is expressed as:

\[

H_\nu\{f(r)\}(k) = \int_0^{\infty} f(r) J_\nu(kr) r \, dr,

\]

where \(k \geq 0\). The inverse Hankel transform reconstructs the original function from its transform, and it shares properties such as linearity and convolution similar to Fourier transforms.

The Hankel transform is extensively utilized in solving partial differential equations with radial symmetry, such as the heat equation, wave equation, and problems involving potential fields in physics. Its application extends to signal processing, image analysis, and acoustics, particularly when radial or cylindrical data are involved.

Mathematically, the Hankel transform provides a tool for converting differential operations into algebraic ones, simplifying the analysis of systems with rotational invariance. Its properties include self-inversion and eigenfunction relations with Bessel functions, making it a valuable component in the set of integral transforms used in scientific and engineering analyses.