# §20.2 Definitions and Periodic Properties

## §20.2(i) Fourier Series

Corresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to .

## §20.2(ii) Periodicity and Quasi-Periodicity

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

20.2.5,

are the lattice points. The theta functions are quasi-periodic on the lattice:

20.2.6
20.2.7
20.2.8
Figure 20.2.1: -plane. Fundamental parallelogram. Left-hand diagram is the rectangular case ( purely imaginary); right-hand diagram is the general case. zeros of , zeros of , zeros of , zeros of .

With

20.2.10
20.2.11
20.2.12
20.2.13
20.2.14

## §20.2(iv) -Zeros

For , the -zeros of , , are , , , respectively.