Fourier–Bessel series

In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions. In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions. Fourier–Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems. The series formed by the Bessel function of the first kind is known as the Schlömilch's Series. The Fourier–Bessel series of a function f(x) with a domain of is the notation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind Jα, where the argument to each version n is differently scaled, according to where uα,n is a root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients: The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ coordinate of cylindrical coordinates. Just as the Fourier series is defined for a finite interval and has a counterpart, the continuous Fourier transform over an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the Hankel transform. As said, differently scaled Bessel Functions are orthogonal with respect to the inner product