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Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation + + = for an arbitrary complex number, which represents the order of the Bessel function.
The angular integration of an exponential in cylindrical coordinates can be written in terms of Bessel functions of the first kind [4] [5]: 113 ( ()) = and ( ()) = (). For applications of these integrals see Magnetic interaction between current loops in a simple plasma or electron gas .
Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld. The function () is the zeroth-order Bessel function of the first kind, better known by the notation () = in English literature. This identity is known as the Sommerfeld identity.
In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin. One such formula is the following integral formula involving a product of three Bessel functions:
The necessary coefficient F ν 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.
In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1906a, 1906b, 1905a, 1905b). The third Jackson q -Bessel function is the same as the Hahn–Exton q -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 Bessel function of the first kind has the power series = = (+ +)! +By Ramanujan's master theorem, together with some identities for the gamma function and rearranging, we can evaluate the integral