§20.2 Definitions and Periodic Properties

§20.2(i) Fourier Series

 20.2.1 $\displaystyle\theta_{1}\left(z\middle|\tau\right)$ $\displaystyle=\theta_{1}\left(z,q\right)=2\sum_{n=0}^{\infty}(-1)^{n}q^{(n+% \frac{1}{2})^{2}}\sin\left((2n+1)z\right),$ 20.2.2 $\displaystyle\theta_{2}\left(z\middle|\tau\right)$ $\displaystyle=\theta_{2}\left(z,q\right)=2\sum_{n=0}^{\infty}q^{(n+\frac{1}{2}% )^{2}}\cos\left((2n+1)z\right),$ 20.2.3 $\displaystyle\theta_{3}\left(z\middle|\tau\right)$ $\displaystyle=\theta_{3}\left(z,q\right)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos% \left(2nz\right),$ 20.2.4 $\displaystyle\theta_{4}\left(z\middle|\tau\right)$ $\displaystyle=\theta_{4}\left(z,q\right)=1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2% }}\cos\left(2nz\right).$

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

§20.2(ii) Periodicity and Quasi-Periodicity

For fixed $\tau$, each $\theta_{j}\left(z\middle|\tau\right)$ is an entire function of $z$ with period $2\pi$; $\theta_{1}\left(z\middle|\tau\right)$ is odd in $z$ and the others are even. For fixed $z$, each of $\ifrac{\theta_{1}\left(z\middle|\tau\right)}{\sin z}$, $\ifrac{\theta_{2}\left(z\middle|\tau\right)}{\cos z}$, $\theta_{3}\left(z\middle|\tau\right)$, and $\theta_{4}\left(z\middle|\tau\right)$ is an analytic function of $\tau$ for $\Im\tau>0$, with a natural boundary $\Im\tau=0$, and correspondingly, an analytic function of $q$ for $\left|q\right|<1$ 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

 20.2.5 $z_{m,n}=(m+n\tau)\pi,$ $m,n\in\mathbb{Z}$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathbb{Z}$: set of all integers, $m$: integer, $n$: integer, $z$: complex and $\tau$: lattice parameter A&S Ref: 16.33.1 (in different notation) Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.2.E5 Encodings: TeX, pMML, png See also: Annotations for §20.2(ii), §20.2 and Ch.20

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

 20.2.6 $\displaystyle\theta_{1}\left(z+(m+n\tau)\pi\middle|\tau\right)$ $\displaystyle=(-1)^{m+n}q^{-n^{2}}e^{-2inz}\theta_{1}\left(z\middle|\tau\right),$ 20.2.7 $\displaystyle\theta_{2}\left(z+(m+n\tau)\pi\middle|\tau\right)$ $\displaystyle=(-1)^{m}q^{-n^{2}}e^{-2inz}\theta_{2}\left(z\middle|\tau\right),$ 20.2.8 $\displaystyle\theta_{3}\left(z+(m+n\tau)\pi\middle|\tau\right)$ $\displaystyle=q^{-n^{2}}e^{-2inz}\theta_{3}\left(z\middle|\tau\right),$ 20.2.9 $\displaystyle\theta_{4}\left(z+(m+n\tau)\pi\middle|\tau\right)$ $\displaystyle=(-1)^{n}q^{-n^{2}}e^{-2inz}\theta_{4}\left(z\middle|\tau\right).$

§20.2(iii) Translation of the Argument by Half-Periods

With

 20.2.10 $M\equiv M(z|\tau)=e^{iz+(i\pi\tau/4)},$ ⓘ Defines: $M$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter A&S Ref: 16.33.1 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E10 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20
 20.2.11 $\displaystyle\theta_{1}\left(z\middle|\tau\right)$ $\displaystyle=-\theta_{2}\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=-iM\theta_% {4}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=-iM\theta_{3}\left(z+\tfrac{% 1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex, $\tau$: lattice parameter and $M$ A&S Ref: 16.33.2 (in different notation) Referenced by: §20.7(iv) Permalink: http://dlmf.nist.gov/20.2.E11 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20 20.2.12 $\displaystyle\theta_{2}\left(z\middle|\tau\right)$ $\displaystyle=\theta_{1}\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=M\theta_{3}% \left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=M\theta_{4}\left(z+\tfrac{1}{2}% \pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex, $\tau$: lattice parameter and $M$ A&S Ref: 16.33.3 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E12 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20 20.2.13 $\displaystyle\theta_{3}\left(z\middle|\tau\right)$ $\displaystyle=\theta_{4}\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=M\theta_{2}% \left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=M\theta_{1}\left(z+\tfrac{1}{2}% \pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex, $\tau$: lattice parameter and $M$ A&S Ref: 16.33.4 (in different notation) Permalink: http://dlmf.nist.gov/20.2.E13 Encodings: TeX, pMML, png See also: Annotations for §20.2(iii), §20.2 and Ch.20 20.2.14 $\displaystyle\theta_{4}\left(z\middle|\tau\right)$ $\displaystyle=\theta_{3}\left(z+\tfrac{1}{2}\pi\middle|\tau\right)=-iM\theta_{% 1}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=iM\theta_{2}\left(z+\tfrac{1}% {2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right).$

§20.2(iv) $z$-Zeros

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