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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 [1] [failed verification] and then generalized by Friedrich Bessel, [citation needed] 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 .
Theory of Incomplete Cylindrical Functions and their Applications. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg. ISBN 978-3-642-65023-9. Cicchetti, R.; Faraone, A. (December 2004). "Incomplete Hankel and Modified Bessel Functions: A Class of Special Functions for Electromagnetics". IEEE Transactions on Antennas and Propagation.
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:
Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
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 .
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