Hankeltranszformáció

The Hankel transform is a mathematical integral transform named after the German mathematician Hermann Hankel. It is closely related to the Fourier transform but is specifically designed for functions of a single variable that depend on the radial distance from the origin in a two-dimensional polar coordinate system. In essence, it transforms a function of the radial coordinate into a function of the angular frequency or radial wave number.

The Hankel transform of order $\nu$ for a function $f(r)$ is defined as $F_\nu(k) = \int_0^\infty r f(r) J_\nu(kr) dr$, where $J_\nu(x)$ is the Bessel function of the first kind of order $\nu$. The inverse Hankel transform allows one to recover the original function $f(r)$ from its transformed version $F_\nu(k)$. The formula for the inverse transform is $f(r) = \int_0^\infty k F_\nu(k) J_\nu(kr) dk$. The order $\nu$ of the Bessel function depends on the specific problem and symmetry considerations.

This transform is particularly useful in solving problems involving circular symmetry in fields like physics and engineering. Applications include solving the wave equation, heat conduction problems, and scattering problems in cylindrical or spherical coordinates. It is also used in signal processing and image analysis where radial symmetry is present. The Hankel transform provides an efficient way to analyze and manipulate such radially symmetric data.