# §21.2 Definitions

## §21.2(i) Riemann Theta Functions

 21.2.1 $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in% {\mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}% }\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}.$ ⓘ Defines: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: 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 $\boldsymbol{{\Omega}}$: 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 $\boldsymbol{{\Omega}}$ spaces; hence $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is an analytic function of (each element of) $\mathbf{z}$ and (each element of) $\boldsymbol{{\Omega}}$. $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ 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 $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, defined by (Deconinck et al. (2004)),

 21.2.2 $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{-\pi[\Im% \mathbf{z}]\cdot[\Im\boldsymbol{{\Omega}}]^{-1}\cdot[\Im\mathbf{z}]}\theta% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$ ⓘ Defines: $\hat{\theta}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: scaled Riemann theta function Symbols: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\Im$: imaginary part and $\boldsymbol{{\Omega}}$: 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

$\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ 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).

### Example

 21.2.3 $\theta\left(z_{1},z_{2}\middle|\begin{bmatrix}i&-\tfrac{1}{2}\\ -\tfrac{1}{2}&i\end{bmatrix}\right)=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=-% \infty}^{\infty}e^{-\pi(n_{1}^{2}+n_{2}^{2})}e^{-i\pi n_{1}n_{2}}e^{2\pi i(n_{% 1}z_{1}+n_{2}z_{2})}.$

With $z_{1}=x_{1}+iy_{1}$, $z_{2}=x_{2}+iy_{2}$,

 21.2.4 $\hat{\theta}\left(x_{1}+iy_{1},x_{2}+iy_{2}\middle|\begin{bmatrix}i&-\tfrac{1}% {2}\\ -\tfrac{1}{2}&i\end{bmatrix}\right)=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=-% \infty}^{\infty}e^{-\pi(n_{1}+y_{1})^{2}-\pi(n_{2}+y_{2})^{2}}\*e^{\pi i(2n_{1% }x_{1}+2n_{2}x_{2}-n_{1}n_{2})}.$

## §21.2(ii) Riemann Theta Functions with Characteristics

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

 21.2.5 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in{% \mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}[\mathbf{n}+\boldsymbol{{\alpha}}]% \cdot\boldsymbol{{\Omega}}\cdot[\mathbf{n}+\boldsymbol{{\alpha}}]+[\mathbf{n}+% \boldsymbol{{\alpha}}]\cdot[\mathbf{z}+\boldsymbol{{\beta}}]\right)}.$ ⓘ Defines: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: 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, $\boldsymbol{{\Omega}}$: a Riemann matrix, $\boldsymbol{{\alpha}}$: $g$-dimensional vector and $\boldsymbol{{\beta}}$: $g$-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 $\begin{bmatrix}\boldsymbol{{\alpha}}\\ \boldsymbol{{\beta}}\end{bmatrix}$. It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor:

 21.2.6 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\frac{1}{2% }\boldsymbol{{\alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+% \boldsymbol{{\alpha}}\cdot[\mathbf{z}+\boldsymbol{{\beta}}]\right)}\theta\left% (\mathbf{z}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}% \middle|\boldsymbol{{\Omega}}\right),$

and

 21.2.7 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{0}}}{\boldsymbol{{0}}}\left(\mathbf% {z}\middle|\boldsymbol{{\Omega}}\right)=\theta\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right).$

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

## §21.2(iii) Relation to Classical Theta Functions

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

 21.2.8 $\theta\left(z\middle|\Omega\right)=\theta_{3}\left(\pi z\middle|\Omega\right),$
 21.2.9 $\displaystyle\theta_{1}\left(\pi z\middle|\Omega\right)$ $\displaystyle=-\theta\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}}{\frac{1}{2}}\left(z% \middle|\Omega\right),$ 21.2.10 $\displaystyle\theta_{2}\left(\pi z\middle|\Omega\right)$ $\displaystyle=\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}}{0}\left(z\middle|% \Omega\right),$ 21.2.11 $\displaystyle\theta_{3}\left(\pi z\middle|\Omega\right)$ $\displaystyle=\theta\genfrac{[}{]}{0.0pt}{}{0}{0}\left(z\middle|\Omega\right),$ 21.2.12 $\displaystyle\theta_{4}\left(\pi z\middle|\Omega\right)$ $\displaystyle=\theta\genfrac{[}{]}{0.0pt}{}{0}{\tfrac{1}{2}}\left(z\middle|% \Omega\right).$