In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r axis. The necessary coefficient of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval. Definition The Hankel transform of order of a function f(r) is given by where is the Bessel function of the first kind of order with . The inverse Hankel transform of is defined as which can be readily verified using the orthogonality relationship described below. Domain of definition Inverting a Hankel transform of a function f(r) is valid at every point at which f(r) is continuous, provided that the function is defined in (0, ∞), is piecewise continuous and of bounded variation in every finite subinterval in (0, ∞), and However, like the Fourier transform, the domain can be extended by a density argument to include some functions whose above integral is not finite, for example . Alternative definition An alternative definition says that the Hankel transform of g(r) is The two definitions are related: If , then This means that, as with the previous definition, the Hankel transform defined this way is also its own inverse: The obvious domain now has the condition but this can be extended. According to the reference given above, we can take the integral as the limit as the upper limit goes to infinity (an improper integral rather than a Lebesgue integral), and in this way the Hankel transform and its inverse work for all functions in L2(0, ∞). Transforming Laplace's equation The Hankel transform can be used to transform and solve Laplace's equation expressed in cylindrical coordinates. Under the Hankel transform, the Bessel operator becomes a multiplication by . In the axisymmetric case, the partial differential equation is transformed as which is an ordinary differential equation in the transformed variable