A sine wave is the canonical periodic function, so an obvious way to create a periodic function of two variables would be to multiply two sine waves:

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

This function is doubly periodic: periodic in the horizontal and vertical directions.

Now suppose you want to construct a doubly periodic function of a complex variable *z* = *x* + *iy*. One thing you might try is

*f*(*x* + *iy*) = sin(*x*) sin(*y*)

This is a function of a complex variable, technically, but it’s not what we usually have in mind when we speak of functions of a complex variable. We’re usually interested in complex functions that are differentiable as complex functions.

If *h* is the increment in the definition of a derivative, we require the limit as *h* approaches 0 to be equal no matter what route *h* takes. On the real line, *h* could go to zero from the left or from the right. That’s all the possibilities. But in the complex plane, *h* could approach the origin from any angle, or it could take a more complex route such as spiraling toward 0.

It turns out that it is sufficient that the limit be the same whether *h* goes to 0 along the real or imaginary axis. This gives us the Cauchy-Riemann equations. If

*f*(*x*, *y*) = *u*(*x*, *y*) + *i* *v*(*x*, *y*)

then the Cauchy-Riemann equations require the partial derivative of *u* with respect to *x* to equal the partial derivative of *v* with respect to *y*, and the partial derivative of *v* with respect to *x* to be the negative of the partial of *u* with respect to *y*.

*u*_{x} = *v*_{y}

*v*_{x} = –*u*_{y}

These equations imply that if *v* = 0 then *u* is constant. A real-valued function of a complex variable cannot be analytic unless its constant.

So can we rescue our example by making up an imaginary component *v*? If so

*f*(*x*, *y*) = sin(*x*) sin(*y*) + *i* *v*(*x*, *y*)

would have to satisfy the Cauchy-Riemann equations. The first equation would require

*v*(*x*, *y*) = – cos(*x*) cos(*y*) + *a* cos(*x*)

for some constant *a*, but the second would require

*v*(*x*, *y*) = cos(*x*) cos(*y*) + *b * cos(*y*)

for some constant *b*. Note the negative sign in the former but not in the latter. I’ll leave it as an exercise for the reader to show that these requirements are contradictory.

Liouville’s theorem says that a bounded analytic function must be constant. If an analytic function *f* is doubly periodic, then it must either be constant or have a singularity. There are many such functions, known as elliptic functions.