Besselfunctions

Bessel functions are a set of solutions to Bessel's differential equation, a second-order linear differential equation. They are important in many areas of physics and engineering, including wave propagation, acoustics, and electromagnetism. The general form of Bessel's differential equation is $x^2 y'' + xy' + (x^2 - \nu^2)y = 0$, where $\nu$ is a constant known as the order of the Bessel function.

There are several types of Bessel functions, with the most common being Bessel functions of the first kind, denoted by $J_\nu(x)$, and Bessel functions of the second kind, denoted by $Y_\nu(x)$. Bessel functions of the first kind are finite at $x=0$, while Bessel functions of the second kind have a singularity at $x=0$. Other related functions include Bessel functions of the third kind (Hankel functions) and modified Bessel functions.

Bessel functions can be defined by their power series expansions. For example, the Bessel function of the first kind of integer order $\nu=n$ is given by the series $J_n(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k!(n+k)!} (\frac{x}{2})^{2k+n}$. These functions exhibit oscillatory behavior and are fundamental to solving problems involving cylindrical symmetry. They are often encountered when solving Laplace's or Helmholtz's equation in cylindrical coordinates. Their properties, such as orthogonality and recurrence relations, make them powerful tools in mathematical physics.