§21.2 Definitions

21.2.1		$$\theta \left(\mathbf{z}\right|\mathbf{\Omega })=\sum _{\mathbf{n}\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}\mathbf{n}\cdot \mathbf{\Omega }\cdot \mathbf{n}+\mathbf{n}\cdot \mathbf{z}\right)}.$$
ⓘ Defines: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $\mathbf{\Omega }$: a Riemann matrix Referenced by: §21.2(ii) Permalink: http://dlmf.nist.gov/21.2.E1 Encodings: TeX, pMML, png See also: Annotations for §21.2(i), §21.2 and Ch.21

This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $\mathbf{z}$ and $\mathbf{\Omega }$ spaces; hence $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$ is an analytic function of (each element of) $\mathbf{z}$ and (each element of) $\mathbf{\Omega }$. $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$ is also referred to as a theta function with $g$ components, a $g$-dimensional theta function or as a genus $g$ theta function.

For numerical purposes we use the scaled Riemann theta function $\widehat{\theta }\left(\mathbf{z}\right|\mathbf{\Omega })$, defined by (Deconinck et al. (2004)),

21.2.2		$$\widehat{\theta }\left(\mathbf{z}\right|\mathbf{\Omega })={\mathrm{e}}^{-\pi [\mathrm{\Im }\mathbf{z}]\cdot {[\mathrm{\Im }\mathbf{\Omega }]}^{-1}\cdot [\mathrm{\Im }\mathbf{z}]}\theta \left(\mathbf{z}\right|\mathbf{\Omega }).$$
ⓘ Defines: $\widehat{\theta }\left(\mathbf{z}\right|\mathbf{\Omega })$: scaled Riemann theta function Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part and $\mathbf{\Omega }$: a Riemann matrix Permalink: http://dlmf.nist.gov/21.2.E2 Encodings: TeX, pMML, png See also: Annotations for §21.2(i), §21.2 and Ch.21

$\widehat{\theta }\left(\mathbf{z}\right|\mathbf{\Omega })$ is a bounded nonanalytic function of $\mathbf{z}$. Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. See also §21.10(i).

Let $\mathit{\alpha },\mathit{\beta }\in {ℝ}^g$. Define

21.2.5		$$\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })=\sum _{\mathbf{n}\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}[\mathbf{n}+\mathit{\alpha }]\cdot \mathbf{\Omega }\cdot [\mathbf{n}+\mathit{\alpha }]+[\mathbf{n}+\mathit{\alpha }]\cdot [\mathbf{z}+\mathit{\beta }]\right)}.$$
ⓘ Defines: $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer, $\mathbf{\Omega }$: a Riemann matrix, $\mathit{\alpha }$: $g$-dimensional vector and $\mathit{\beta }$: $g$-dimensional vector Permalink: http://dlmf.nist.gov/21.2.E5 Encodings: TeX, pMML, png See also: Annotations for §21.2(ii), §21.2 and Ch.21

This function is referred to as a Riemann theta function with characteristics $\left[\begin{array}{c}\hfill \mathit{\alpha }\hfill \\ \hfill \mathit{\beta }\hfill \end{array}\right]$. It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor:

21.2.6		$$\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })={\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}\mathit{\alpha }\cdot \mathbf{\Omega }\cdot \mathit{\alpha }+\mathit{\alpha }\cdot [\mathbf{z}+\mathit{\beta }]\right)}\theta \left(\mathbf{z}+\mathbf{\Omega }\mathit{\alpha }+\mathit{\beta }\right|\mathbf{\Omega }),$$
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbf{\Omega }$: a Riemann matrix, $\mathit{\alpha }$: $g$-dimensional vector and $\mathit{\beta }$: $g$-dimensional vector Permalink: http://dlmf.nist.gov/21.2.E6 Encodings: TeX, pMML, png See also: Annotations for §21.2(ii), §21.2 and Ch.21

and

21.2.7		$$\theta \left[\genfrac{}{}{0.0pt}{}{\mathbf{0}}{\mathbf{0}}\right]\left(\mathbf{z}\right|\mathbf{\Omega })=\theta \left(\mathbf{z}\right|\mathbf{\Omega }).$$
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics and $\mathbf{\Omega }$: a Riemann matrix Permalink: http://dlmf.nist.gov/21.2.E7 Encodings: TeX, pMML, png See also: Annotations for §21.2(ii), §21.2 and Ch.21

Characteristics whose elements are either $0$ or $\frac{1}{2}$ are called half-period characteristics. For given $\mathbf{\Omega }$, there are $2^{2g}$ $g$-dimensional Riemann theta functions with half-period characteristics.

For $g=1$, and with the notation of §20.2(i),

21.2.8		$$\theta \left(z\right|\mathrm{\Omega })={\theta }_3\left(\pi z\right|\mathrm{\Omega }),$$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/21.2.E8 Encodings: TeX, pMML, png See also: Annotations for §21.2(iii), §21.2 and Ch.21

21.2.9		${\theta }_1\left(\pi z\right|\mathrm{\Omega })$	$=-\theta \left[{\displaystyle \genfrac{}{}{0.0pt}{}{\frac{1}{2}}{\frac{1}{2}}}\right]\left(z\right|\mathrm{\Omega }),$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/21.2.E9 Encodings: TeX, pMML, png See also: Annotations for §21.2(iii), §21.2 and Ch.21
21.2.10		${\theta }_2\left(\pi z\right|\mathrm{\Omega })$	$=\theta \left[{\displaystyle \genfrac{}{}{0.0pt}{}{\frac{1}{2}}{0}}\right]\left(z\right|\mathrm{\Omega }),$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/21.2.E10 Encodings: TeX, pMML, png See also: Annotations for §21.2(iii), §21.2 and Ch.21
21.2.11		${\theta }_3\left(\pi z\right|\mathrm{\Omega })$	$=\theta \left[{\displaystyle \genfrac{}{}{0.0pt}{}{0}{0}}\right]\left(z\right|\mathrm{\Omega }),$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/21.2.E11 Encodings: TeX, pMML, png See also: Annotations for §21.2(iii), §21.2 and Ch.21
21.2.12		${\theta }_4\left(\pi z\right|\mathrm{\Omega })$	$=\theta \left[{\displaystyle \genfrac{}{}{0.0pt}{}{0}{\frac{1}{2}}}\right]\left(z\right|\mathrm{\Omega }).$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter Permalink: http://dlmf.nist.gov/21.2.E12 Encodings: TeX, pMML, png See also: Annotations for §21.2(iii), §21.2 and Ch.21