Liouvillian function

Definition

Let $K$ be a differential field 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.