Bessel function

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2025) (Learn how and when to remove this message)

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 $${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}$$ for an arbitrary complex number ${\displaystyle \alpha }$, which represents the order of the Bessel function. Although ${\displaystyle \alpha }$ and ${\displaystyle -\alpha }$ produce the same differential equation, 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 ${\displaystyle \alpha }$.

The most important cases are when ${\displaystyle \alpha }$ is an integer or half-integer. Bessel functions for integer ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha }$ are obtained when solving the Helmholtz equation in spherical coordinates.