---
tag: 87DAYKCZ5M
kind: concept
name: Heun function
msc: ["33E30", 34M03, 34M35]
sources:
  - url: https://en.wikipedia.org/wiki/Heun_function
same_as:
  wikidata: Q5748204
  wikipedia: Heun function
---

## Summary

Heun functions are solutions of Heun's equation, the general second order linear [[Fuchsian equation|ZBVH2YE50B]] with four [[regular singular points|H96HXG6CA0]], 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|W8VE25R5A7]], which has three regular singular points.
It contains the Lamé, Mathieu and spheroidal wave equations as special or confluent cases.

## Explanations

### L1 human

The hypergeometric equation is fixed, up to its parameters, by the positions of its three singular points and the exponents there.
With four singular points this is no longer true: besides the exponents there is one extra number $q$, the accessory parameter.
This extra freedom makes Heun functions much harder to handle than hypergeometric functions.
They come up in quantum mechanics and in general relativity, among other places.
A trivial example: when $q = 0$ and $\alpha = 0$, the constant function $1$ solves Heun's equation.