### John O'Hare

Shared publicly -**Elliptic functions**

You probably know about trig functions like sin(x). These are the most basic functions that are periodic:

sin(x + 2π) = sin(x)

**Elliptic functions**are functions of two variables, x and y, that are periodic in two directions:

f(x + 2π,y) = f(x,y)

and

f(x,y + 2π) = f(x,y)

This movie is a way of illustrating an elliptic function.

What makes elliptic functions so special is that you can think of them as functions of a single complex variable:

z = x + iy

and then they have a derivative in the special sense you learn about in a course on complex functions!

It's a lot harder for a complex function to have a derivative than an ordinary real function! A function like

f(x,y) = sin(x) sin(y)

is periodic in two directions, but it doesn't have a derivative df/dz. Mysterious as this may sound, this is the reason elliptic functions are so special.

In the late 1800s, all the best mathematicians thought about elliptic functions, so there are 'Jacobi elliptic functions' and 'Weierstrass elliptic functions' and many more. Now they're less popular, but they're still incredibly important. You need to think about them if you want to deeply understand how long the perimeter of an ellipse is. They're also important in physics, and fundamental to the proof of Fermat's Last Theorem.

+Gerard Westendorp has been making mathematical illustrations for a long time, so if you like such things, circle him!

A Jacobi elliptic function. This time The checkerboard pattern is animated only over magnitude, not phase, so that the squares appear to originate from… - Gerard Westendorp - Google+

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