---
tag: N5PM2NCPWF
kind: concept
name: Liouvillian function
msc: [12H05, 34M15]
sources:
  - url: https://en.wikipedia.org/wiki/Liouvillian_function
same_as:
  wikidata: Q25303953
  wikipedia: Liouvillian function
---

## Summary

A Liouvillian function is built from rational functions in finitely many steps by taking algebraic functions, integrals and exponentials of integrals.
They make precise the idea of a solution "in closed form" for differential equations.
[[Differential Galois theory|EEZS4H8QPH]] decides whether a linear differential equation has Liouvillian solutions.
See the [[definition|R80FA7993W]].

## Explanations

### L1 human

Start with fractions of polynomials in $x$.
Allow three operations again and again: solving polynomial equations, taking antiderivatives, and taking exponentials of antiderivatives.
Everything one can reach is Liouvillian; for example $e^{x^2}$ and the error function $\int e^{-x^2} dx$ are.
Elementary functions such as $\sin x$, $\ln x$ and $\sqrt{1 + x^3}$ are all Liouvillian.
The solutions of the [[Airy equation|PNESAD6R5T]] $y'' = x y$ are not.