# Doubly periodic function

## From Wikipedia, the free encyclopedia

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

- $f(z+u)=f(z+v)=f(z)\,$

for all values of the complex number *z*.^{[1]}^{[2]}

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 *e*^{z} is a singly periodic function, with period 2*πi*.