Doubly periodic function

In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers ${\displaystyle u}$ and ${\displaystyle v}$ that are linearly independent as vectors over the field of real numbers. That ${\displaystyle u}$ and ${\displaystyle v}$ are periods of a function ${\displaystyle f}$ means that

${\displaystyle f(z+u)=f(z+v)=f(z)\,}$

for all values of the complex number ${\displaystyle z}$.

The doubly periodic function is thus a two-dimensional extension of the simpler singly periodic function, which repeats itself in a single dimension. Familiar examples of functions with a single period on the real number line include the trigonometric functions like cosine and sine, In the complex plane the exponential function ${\displaystyle e^{z}}$ is a singly periodic function, with period ${\displaystyle 2\pi i}$.