§20.2 Definitions and Periodic Properties

Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$.

For fixed $\tau$, each ${\theta }_j\left(z\right|\tau )$ is an entire function of $z$ with period $2\pi$; ${\theta }_1\left(z\right|\tau )$ is odd in $z$ and the others are even. For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$.

The four points $(0,\pi ,\pi +\tau \pi ,\tau \pi )$ are the vertices of the fundamental parallelogram in the $z$-plane; see Figure 20.2.1. The points

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

With

For $m,n\in \mathbb{Z}$, the $z$-zeros of ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, are $(m+n\tau )\pi$, $(m+\frac{1}{2}+n\tau )\pi$, $(m+\frac{1}{2}+(n+\frac{1}{2})\tau )\pi$, $(m+(n+\frac{1}{2})\tau )\pi$ respectively.