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
.
On the other hand, trying something like
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
and
so that the lattice consists of the assemblage
wherein *m* and *n* range over all the signed integers. As a matter of notation, it is convenient to use
when the origin (0,0) has been dropped from .

- Eisenstein series
- Weierstrass function
- restraints on periodic functions
- the differential equation for
- the Jacobi functions sn, cn, and dn