Heun function

Summary

Heun functions are solutions of Heun's equation, the general second order linear Fuchsian equation with four regular singular points, placed at $0, 1, a$ and $\infty$. The local Heun function is the solution that is analytic at $0$ with value $1$. Heun's equation is the next case after the hypergeometric differential equation, which has three regular singular points. It contains the Lamé, Mathieu and spheroidal wave equations as special or confluent cases.