# §21.3 Symmetry and Quasi-Periodicity

## §21.3(i) Riemann Theta Functions

 21.3.1 $\theta\left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\theta\left(% \mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$ ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: Riemann theta function and $\boldsymbol{{\Omega}}$: a Riemann matrix Permalink: http://dlmf.nist.gov/21.3.E1 Encodings: TeX, pMML, png See also: Annotations for §21.3(i), §21.3 and Ch.21
 21.3.2 $\theta\left(\mathbf{z}+\mathbf{m}_{1}\middle|\boldsymbol{{\Omega}}\right)=% \theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$ ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: Riemann theta function and $\boldsymbol{{\Omega}}$: a Riemann matrix Permalink: http://dlmf.nist.gov/21.3.E2 Encodings: TeX, pMML, png See also: Annotations for §21.3(i), §21.3 and Ch.21

when $\mathbf{m}_{1}\in{\mathbb{Z}^{g}}.$ Thus $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is periodic, with period $1$, in each element of $\mathbf{z}$. More generally,

 21.3.3 $\theta\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}% \middle|\boldsymbol{{\Omega}}\right)=e^{-2\pi i\left(\frac{1}{2}\mathbf{m}_{2}% \cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}+\mathbf{m}_{2}\cdot\mathbf{z}% \right)}\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$

with $\mathbf{m}_{1}$, $\mathbf{m}_{2}$ $\in{\mathbb{Z}^{g}}$. This is the quasi-periodicity property of the Riemann theta function. It determines the Riemann theta function up to a constant factor. The set of points $\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}$ form a $g$-dimensional lattice, the period lattice of the Riemann theta function.

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

Again, with $\mathbf{m}_{1}$, $\mathbf{m}_{2}$ $\in{\mathbb{Z}^{g}}$

 21.3.4 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right)=e^{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}\theta% \genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(% \mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$ ⓘ Symbols: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\boldsymbol{{\Omega}}$: a Riemann matrix, $\boldsymbol{{\alpha}}$: $g$-dimensional vector and $\boldsymbol{{\beta}}$: $g$-dimensional vector Referenced by: item Equation (21.3.4) Permalink: http://dlmf.nist.gov/21.3.E4 Encodings: TeX, pMML, png Errata (effective with 1.0.5): Originally the vector $\mathbf{m}_{2}$ on the right-hand-side was given incorrectly as $\mathbf{m}_{1}$. Reported 2012-08-27 by Klaas Vantournhout See also: Annotations for §21.3(ii), §21.3 and Ch.21

Because of this property, the elements of $\boldsymbol{{\alpha}}$ and $\boldsymbol{{\beta}}$ are usually restricted to $[0,1)$, without loss of generality.

 21.3.5 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}\middle|% \boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\boldsymbol{{\alpha}}\cdot\mathbf{% m}_{1}-\boldsymbol{{\beta}}\cdot\mathbf{m}_{2}-\frac{1}{2}\mathbf{m}_{2}\cdot% \boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}-\mathbf{m}_{2}\cdot\mathbf{z}\right)}% \theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

For Riemann theta functions with half-period characteristics,

 21.3.6 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=(-1)^{4\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\beta}}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{% \alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).$

See also §20.2(iii) for the case $g=1$ and classical theta functions.