Functions like sine and cosine are periodic. For example, sin(x + 2πn) = sin(x) for all x and any integer n, and so the period of sine is 2π. But what happens if you look at sine or cosine as functions of a complex variable? They’re still periodic if you shift left or right, but not if you shift up or down. If you move up or down, i.e. in a pure imaginary direction, sines and cosines become unbounded.
Doubly periodic functions are periodic in two directions. Formally, a function f(z) of complex variable is doubly periodic if there exist two constants ω1 and ω2 such that
f(z) = f(z + ω1) = f(z + ω2)
for all z. The ratio ω1 / ω2 cannot be real; otherwise the two periods point in the same direction. For the rest of this post, I’ll assume ω1 = 1 and ω2 = i. Such functions are periodic in the horizontal (real-axis) and vertical (imaginary-axis) directions. They repeat everywhere their behavior on the unit square.
What do doubly periodic functions look like? It depends on what restrictions we place on the functions. When we’re working with complex functions, we’re typically interested in functions that are analytic, i.e. differentiable as complex functions.
Only constant functions can be doubly periodic and analytic everywhere. Why? Our functions take on over and over the values they take on over the unit square. If a function is analytic over the (closed) unit square then it’s bounded over that square, and since it’s doubly periodic, it’s bounded everywhere. By Liouville’s theorem, the only bounded analytic functions are constants.
This says that to find interesting doubly periodic functions, we’ll have to relax our requirements. Instead of requiring functions to be analytic everywhere, we will require them to be analytic except at isolated singularities. That is, the functions are allowed to blow up at a finite number of points. There’s a rich set of such functions, known as elliptic functions.
There are two well-known families of elliptic functions. One is the Weierstrass ℘ function (TeX symbol
wp, Unicode U+2118) and its derivatives. The other is the Jacobi functions sn, cn, and dn. These functions have names resembling familiar trig functions because the Jacobi functions are in some ways analogous to trig functions.
It turns out that all elliptic functions can be written as combinations either of the Weierstrass function and its derivative or combinations of Jacobi functions. Roughly speaking, Weierstrass functions are easier to work with theoretically and Jacobi functions are easier to work with numerically.
Related: Applied complex analysis