---
tag: HWY72ZK273
kind: concept
name: Special functions
msc: ["33-00", "33E99"]
sources:
  - url: https://en.wikipedia.org/wiki/Special_functions
same_as:
  wikidata: Q867707
  wikipedia: Special functions
---

## Summary

Special functions are the named functions of analysis that occur again and again in mathematics and physics, beyond the elementary functions.
Examples are the [[gamma function|0H5SYBSED9]], the [[hypergeometric function|463X5V31D9]], the [[Bessel functions|CZCMRDWDYW]], the [[Legendre polynomials|QFYGJVA996]] and the [[elliptic integrals|NXMCYGCB59]].
The term has no formal definition: a function counts as special when it has an established name, notation and theory.
Their study collects identities, integral representations, series, asymptotic formulas and differential equations that make these functions usable in computation.

## Explanations

### L1 human

Polynomials, exponentials, logarithms and trigonometric functions are the everyday functions of school mathematics.
Many problems lead to functions that are not of this kind, but appear so often that they get a name, like a new word in a language.
For example, the area under the bell curve $e^{-x^2}$ from $0$ to $x$ cannot be written with elementary functions, so it gets its own name, the error function.
Tables, formulas and computer libraries then treat these functions as well as the familiar ones.
Many special functions solve a simple differential equation or extend a sequence such as the factorials to non-integer arguments.