Bessel function

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