Shows up when trying to solve 2D wave equation on a circular domain in polar coordinates with separation of variables, where we have to decompose the initial condition in termes of a fourier-Bessel series, exactly like the Fourier series appears when solving the wave equation in linear coordinates.
For the same fundamental reasons, also appears when calculating the Schrödinger equation solution for the hydrogen atom.
Completeness: math.stackexchange.com/questions/2192665/is-this-set-of-bessel-functions-a-basis-for-all-c10-a-functions TODO
This is the Bessel function analogue to Fourier basis is complete for .
Articles by others on the same topic
Bessel functions are standard solutions of the ordinary differential equationThe parameter is the order. A first-kind solution iswhere is the gamma function and a branch of a multivalued function is chosen for a noninteger power. For nonnegative integer , is entire; a second-kind solution is singular at zero.[ref]
These functions arise when separation of variables is applied to the Helmholtz equation in cylindrical coordinates. A circular membrane requires a radial solution regular at the centre, selecting ; its boundary condition then selects zeros of . For example, is the radial profile for the rotationally symmetric family.
Bessel functions are a family of solutions to Bessel's differential equation, which arises in various problems in mathematical physics, particularly in wave propagation, heat conduction, and static potentials. The equation is typically expressed as: \[ x^2 y'' + x y' + (x^2 - n^2) y = 0 \] where \( n \) is a constant, and \( y \) is the function of \( x \).