Elliptic function

Remarks

Then $f(z + \omega) = f(z)$ for every $\omega \in \Lambda$, so $f$ is periodic with each lattice vector as a period. A fundamental parallelogram is a set $\set{a + s\omega_1 + t\omega_2 : 0 \le s, t < 1}$ with $a \in \C$; an elliptic function is determined by its values there. Example: the Weierstrass elliptic function $\wp$ of $\Lambda$ is elliptic with respect to $\Lambda$, and so is its derivative $\wp'$. Constant functions are elliptic for every lattice. Non-example: $e^z$ is meromorphic and periodic, but its periods are exactly the numbers $2\pi i k$ with $k \in \Z$. They all lie on one line through $0$, so there is no second period independent over $\R$. Equivalently, the elliptic functions for $\Lambda$ are the meromorphic functions on the compact Riemann surface $\C / \Lambda$.