What is "bessel function"

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

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

(known as Bessel's differential equation) for an arbitrary complex number , the order of the Bessel function. Although and − produce the same differential equation for real , 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 .

The most important cases are for an integer or half-integer. Bessel functions for integer 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 are obtained when the Helmholtz equation is solved in spherical coordinates.