# Definition:Doubly Periodic Function

Let $f: \C \to \C$ be a complex function.
Then $f \left({z}\right)$ is a doubly-periodic function if there exist $\omega_1, \omega_2 \in \C$ such that:
$(1): \quad \omega_1, \omega_2 \ne 0$
$(2): \quad \dfrac {\omega_1} {\omega_2} \notin \R$
$(3): \quad \forall z \in \C: f \left({z}\right) = f \left({z + \omega_1}\right) = f \left({z + \omega_2}\right)$