Functions like sine and cosine are periodic. For example, sin(*x* + 2π*n*) = 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 ω_{1} and ω_{2} such that

f(*z*) = f(*z* + ω_{1}) = f(*z* + ω_{2})

for all *z*. The ratio ω_{1} / ω_{2} cannot be real; otherwise the two periods point in the same direction. For the rest of this post, I’ll assume ω_{1} = 1 and ω_{2} = 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.

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“we typically are interested in functions that are analytic, i.e. differentiable as complex functions”

Is this the right definition of analytic functions? At least according to Wikipedia (http://bit.ly/mb5zVO):

“A function is analytic if and only if it is equal to its Taylor series in some neighborhood of every point.”

“As noted above, any analytic function (real or complex) is infinitely differentiable … There exist smooth real functions which are not analytic”

One of the amazing theorems of complex analysis is that if a function is differentiable once, it’s infinitely differentiable and analytic. This is very different from the real setting where a function can be differentiable once but not twice, or as you point out, infinitely differentiable without being analytic.

How can this be? Look at the definition of derivative, the limit of ( f(z + h) – f(z) ) / h. When h is complex, h can go to zero from any angle. Requiring the limit to be the same from every direction is a severe restriction. So having a complex derivative is a much stronger requirement than having real partial derivatives.

Thanks for the extra clarification!

We should mention the connection with elliptic curve cryptography. The Weierstrass function and its derivative satisfy an equation like y^2=x^3+ax+b, and as a plane cubic this has a group law. The analytic discovery of the addition formula is difficult, but the algebraic version is rather easy. Once you have the group law you can mimic the Diffie-Hellman construction and others.

[Sorry, "let's not forget the connection with analysis" is one of my soapboxes!]

Thanks for bringing up elliptic curves.

There are a lot of things in math called “elliptic” and the connections between them are not obvious. If I were a man of leisure, I’d like to write a book that explains the connections between ellipses, elliptic functions, elliptic curves, elliptic PDEs, etc.

In fact, the group structure of the elliptic curve (over the complex numbers) equals C modulo period lattice. The isomorphism between this torus and the plane model of the elliptic curve is essentially given by mapping a point from the torus by the Weierstrass-p-function and its derivative.

When you say the functions are allowed to blow up at a finite number of points do you mean a countable number of points? Or perhaps I am misunderstanding something.

I think he means a finite number of points inside a fundamental parallelotope. Or, equivalently, a finite number of points on the resulting torus.

The set of poles is discrete in the complex numbers.

Steven and Felix: Yes, I meant a finite number of singularities in a fundamental region, which turns into a countably infinite number across the complex plane.

A nit…. from what is stated “sin(x + 2πn) = sin(x) for all x and any integer n, and so the period of sine is 2π” doesn’t necessarily follow. All one can say for sure is that 2π is an integral multiple of the period.