Periodic functions

Given a general familiarity with the periodicity of trigonometric functions it is narural to look for periodic functions of a complex variable. However, the trigonometric functions themselves won't do because they turn into hyperbolic functions when given an imaginary argument; for example $\cos(i\theta) = \cosh(\theta)$. On the other hand, trying something like $\sin(x)\sin(y)$ readily exhibits two-dimensional periodicity, but trying to combine x and y into the complex variable z = x + i y runs afoul of the Cauchy-Riemann equations, which reveal that such a product is not analytic.

Those two examples are only hasty attempts at guessing functions that could be periodic in the complex plane, whereas a more successful approach would be to start with a periodic configuration like a square grid, followed by taking averages over the grid.

Actually any two vectors will suffice to establish a grid, as long as they are not parallel. By tradition they are called $2\omega$ and $2\omega'$ so that the lattice consists of the assemblage $\Omega = \{m\omega + n\omega'\}$ wherein m and n range over all the signed integers. As a matter of notation, it is convenient to use $\Omega'$ when the origin (0,0) has been dropped from $\Omega$.