3 MARCH 2017 Basics of Bessel function. https://en.m.wikipedia.org/wiki/Bessel_function.

3 MARCH 2017 Basics of Bessel function.
https://en.m.wikipedia.org/wiki/Bessel_function

Bessel function

Bessel functions are the radial part of the modes of vibration of a circular drum.

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of the differential equation

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-\alpha ^{2})y=0}$

(known as Bessel's differential equation) for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation for real α, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.
The most important cases are for α an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates.

Contents

Applications of Bessel functions