Let us begin with some (standard, I think) definitions.

**Def:** An *elliptic function* is a doubly periodic meromorphic function on $\mathbb{C}$.

**Def:** An *elliptic integral* is an integral of the form
$$f(x) = \int_{a}^x R\left(t,\sqrt{P(t)}\right)\ dt,$$
where $R$ is a rational fucntion of its arguments and where $P(t)$ is a third or fourth degree polynomial with simple roots.

I have often heard the claim that an elliptic function is (or can be) defined as the inverse of an elliptic integral. However, I have never seen a proof of this statement. As someone who is largely unfamiliar with the subject, most of the references I could dig up seem to refer to the special case of the Jacobi elliptic functions, which appear as inverse functions of the elliptic integrals of the first kind. Maybe the claim I'm referring to is simply talking about the special case of Jacobi elliptic functions, but I believe the statement holds in generality (I could be wrong).

So, can anyone provide a proof or reference (or counter-example) to something akin to the following?

**Claim:** The elliptic functions are precisely the inverses of the elliptic integrals, as I've defined them above. That is, every elliptic function arises as the inverse of some elliptic integral, and conversely every elliptic integral arises as the inverse of some elliptic function.