Bessel function

Bessel functions are mathematical special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena with circular symmetry or cylindrical symmetry. They are named after the German astronomer and mathematician Friedrich Bessel, who studied them systematically in 1824.^[1]

Bessel functions are solutions to a particular type of ordinary differential equation: $${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0,}$$ where ${\displaystyle \alpha }$ is a number that determines the shape of the solution. This number is called the order of the Bessel function and can be any complex number. Although the same equation arises for both ${\displaystyle \alpha }$ and ${\displaystyle -\alpha }$, mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.

The most important cases are when ${\displaystyle \alpha }$ is an integer or a half-integer. When ${\displaystyle \alpha }$ is an integer, the resulting Bessel functions are often called cylinder functions or cylindrical harmonics because they naturally arise when solving problems (like Laplace's equation) in cylindrical coordinates. When ${\displaystyle \alpha }$ is a half-integer, the solutions are called spherical Bessel functions and are used in spherical systems, such as in solving the Helmholtz equation in spherical coordinates.