---
tag: R80FA7993W
kind: definition
name: Liouvillian function
msc: [12H05, 34M15]
defines: N5PM2NCPWF
uses: [T43TQ817XC]
sources:
  - url: https://en.wikipedia.org/wiki/Liouvillian_function
---

## Definition

Let $K$ be a [[differential field|T43TQ817XC]] of characteristic $0$ with derivation $a \mapsto a'$.
Let $L \supseteq K$ be a differential field extension with the same field of constants as $K$.
It is *Liouvillian* if there is a tower of differential fields $K = K_0 \subseteq \dots \subseteq K_m = L$ with the following property.
For each $i$, $K_i = K_{i-1}(t_i)$, where $t_i$ satisfies one of these conditions:

1. $t_i$ is algebraic over $K_{i-1}$;
2. $t_i' \in K_{i-1}$, so $t_i$ is an integral of an element of $K_{i-1}$;
3. $t_i \ne 0$ and $t_i'/t_i \in K_{i-1}$, so $t_i$ is an exponential of an integral of an element of $K_{i-1}$.

Let $\C(x)$ be the field of rational functions with the derivation $d/dx$.
A *Liouvillian function* is an element of a Liouvillian extension of $\C(x)$, realized as a field of meromorphic functions on some domain.

## Remarks

Example: $t = e^{x^2}$ is Liouvillian over $\C(x)$, since $t'/t = 2x \in \C(x)$ (clause 3).
An antiderivative of $e^{-x^2}$ is Liouvillian too: adjoin $e^{-x^2}$ by clause 3 and then the integral by clause 2.
Also $\ln x$ is Liouvillian by clause 2, since $(\ln x)' = 1/x$, and $\sin x = (e^{ix} - e^{-ix})/(2i)$ is Liouvillian.
Non-example: no nonzero solution of the Airy equation $y'' = x y$ is Liouvillian; this is a result of differential Galois theory, since the differential Galois group of that equation is $\mathrm{SL}_2(\C)$.
Elementary functions are the special case in which clause 2 is restricted to logarithms and clause 3 to exponentials.
So every elementary function is Liouvillian, but $\int e^{-x^2} dx$ is Liouvillian and not elementary, by [[Liouville's theorem|8VRTPECBPZ]].