In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse.

Important elliptic functions are Jacobi elliptic functions and the Weierstrass ${\displaystyle \wp }$-function.

Further development of this theory led to hyperelliptic functions and modular forms.

A meromorphic function is called an elliptic function, if there are two ${\displaystyle \mathbb {R} }$-linear independent complex numbers ${\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }$ such that

So elliptic functions have two periods and are therefore doubly periodic functions.

If ${\displaystyle f}$ is an elliptic function with periods ${\displaystyle \omega _{1},\omega _{2}}$ it also holds that

for every linear combination ${\displaystyle \gamma =m\omega _{1}+n\omega _{2}}$ with ${\displaystyle m,n\in \mathbb {Z} }$.

The abelian group