Liouville's Theorem on integration in finite terms for ${\mathrm{D}}_{\mathrm{\infty }},$ ${\mathrm{S}\mathrm{L}}_2$ and Weierstrass field extensions

Let $k$ be a differential field of characteristic zero and the field of constants $C$ of $k$ be an algebraically closed field. Let $E$ be a differential field extension of $k$ having $C$ as its field of constants and that $E=E_m\supseteq E_{m-1}\supseteq \cdots \supseteq E_1\supseteq E_0=k,$ where $E_i$ is either an elementary extension of $E_{i-1}$ or $E_i=E_{i-1}(t_i,t_i^{\mathrm{\prime }})$ and $t_i$ is weierstrassian (in the sense of Kolchin ([Page 803, Kolchin1953]) over $E_{i-1}$ or $E_i$ is a Picard-Vessiot extension of $E_{i-1}$ having a differential Galois group isomorphic to either the special linear group ${\mathrm{S}\mathrm{L}}_2(C)$ or the infinite dihedral subgroup ${\mathrm{D}}_{\mathrm{\infty }}$ of ${\mathrm{S}\mathrm{L}}_2(C)\mathrm{.}$ In this article, we prove that Liouville’s theorem on integration in finite terms ([Theorem, Rosenlicht1968]) holds for $E$. That is, if $\eta \in E$ and ${\eta }^{\mathrm{\prime }}\in k$ then there is a positive integer $n$ and for $i=1,2,\dots ,n,$ there are elements $c_i\in C,$ $u_i\in k\setminus \{0\}$ and $v\in k$ such that $${\eta }^{\mathrm{\prime }}=\sum _{i=1}^n{c}_i\frac{u_i^{\mathrm{\prime }}}{u_i}+v^{\mathrm{\prime }}\mathrm{.}$$