For fixed , each is an entire function of with period ; is odd in and the others are even. For fixed , each of , , , and is an analytic function of for , with a natural boundary , and correspondingly, an analytic function of for with a natural boundary .
The four points are the vertices of the fundamental parallelogram in the -plane; see Figure 20.2.1. The points
are the lattice points. The theta functions are quasi-periodic on the lattice:
For , the -zeros of , , are , , , respectively.