Double period except for poles

I'm trying to solve a problem in Complex Analysis whose function $f$ is defined in $\mathbb{C}$, is meromorphic and have double period $(f(z)=f(z+a)=f(z+b),\ \frac{a}{b} \notin \mathbb{R})$ except for poles. Is there any relationship between such functions and elliptic functions (where the double period is for all $z \in \mathbb{C}$)?

The fragment "except for poles" is just an indication that the author has reservations to write $f(z_0) = f(z_0+a) = f(z_0+b)$ when $z_0$ is a pole of $f$.
If you view the functions as having values in $\mathbb{C}$, then the poles don't belong to the domain, and hence the above equation would not make sense. If you view your meromorphic functions to have values in the Riemann sphere $\widehat{\mathbb{C}}$, then the poles belong to the domain, and all difficulties nicely disappear.