Are Jacobi elliptic functions liouvillian? I mean: could a Jacobi elliptic function be expressed in terms of a "finite number of arithmetic operations (+ – × ÷), exponentials, constants, solutions of
algebraic equations (a generalization of nth roots), and indefinite integrals of such elements"?

I think I have a proof that the non-constant elliptic functions are not Liouvillian (be careful that the elliptic functions are the inverse of the elliptic integrals).

*

*The Liouvillian functions $L$ is a set defined recursively :

$f(z)  = 1$ and $g(z)  =z$ are in $L$.
If $f(z),g(z)$ are in $L$ then

*

*$a f(z)+b g(z)$, $f(z)g(z)$ are in $L$


*if $f(z)$ is non-constant then $\exp f(z)$, $\log f(z), 1/f(z)$ are in $L$


*$\int f(z)dz$ is in $L$


*$h(z)$ is in $L$ if $\sum_{n=0}^d a_n h(z)^n $ is in $L$



*Then Let $M$ be the constants plus the set of almost everywhere analytic functions such that for every sector $$S_{a,\theta,b} = \{z \in \mathbb{C}, |z-a| >0,  \text{arg}(e^{i \theta} (z-a)) \in (-b,b)\}$$ there is a sub-sector $S_{a',\theta',b'}\subseteq S_{a,\theta,b}$ such that : $f(z)$ is analytic and injective on $S_{a',\theta',b'}$.
From this you can show by induction that if $f(z) \in L$ then $f(z) \in M$,
and since the elliptic functions (the doubly periodic meromorphic functions that are non-constant) are not in $M$, they are not in $L$ neither.

*

*If $f(z) \in M$ and non-constant then $1/f(z),\exp f(z),\log f(z)$  are in $M$.


*If $f(z) \not\in M$ then $\sum_{n=0}^d c_n f(z)^n \not\in M$


*If $f(z),g(z) \in M$ then $ af(z)+bg(z), f(z)g(z)$ are in $M$


*A little work shows that if $f(z) \in M$ then  $\int f(z)dz \in M$
