Definition:Elliptic Function

Definition
Let $\displaystyle y \left({x}\right) = \int_0^x \dfrac {\mathrm d t} {\sqrt {P \left({t}\right)} }$ be an elliptic integral, where $P \left({t}\right)$ is a polynomial of degree $3$ or $4$.

Consider the inverse of $y \left({x}\right)$:
 * $x = \phi \left({y}\right)$

Then $\phi$ is an elliptic function.

Also see

 * Meromorphic Function is Elliptic iff Doubly Periodic