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Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, [1] are canonical solutions y(x) of Bessel's differential equation + + = for an arbitrary complex number, which represents the order of the Bessel function.
The Fourier–Bessel series of a function f(x) with a domain of [0, b] satisfying f(b) = 0. Bessel function for (i) = and (ii) =.: [,] is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind J α, where the argument to each version n is differently scaled, according to [1] [2] ():= (,) where u α,n is a root ...
The von Mises probability density function for the angle x is given by: [2] (,) = ( ()) ()where I 0 is the modified Bessel function of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity: () = ().
where K ν (z) is the ν th order modified Bessel function of the second kind. These functions are named after William Thomson, 1st Baron Kelvin. While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments xe iφ, 0 ≤ φ ...
It was originally developed to compute tables of the modified Bessel function [2] but also applies to Bessel functions of the first kind and has other applications such as computation of the coefficients of Chebyshev expansions of other special functions. [3]
In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case =, are a sequence of polynomials in / used to expand functions in term of Bessel functions. [1] The first few polynomials are () =,
The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name ... is a modified Bessel function of the second kind, y ...
For the special case when μ 1 = μ 2, an asymptotic expansion of the modified Bessel function of the first kind yields for large μ: (; ...