§21.2(i) Riemann Theta Functions

 21.2.1 $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right)=\sum_{\mathbf{n}\in\Integer^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}% \cdot\boldsymbol{{\Omega}}\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}.$ Defines: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$: Riemann theta function Symbols: $\in$: element of, $e$: base of exponential function, $\Integer$: 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

This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $\mathbf{z}$ and $\boldsymbol{{\Omega}}$ spaces; hence $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is an analytic function of (each element of) $\mathbf{z}$ and (each element of) $\boldsymbol{{\Omega}}$. $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ 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 $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$, defined by (Deconinck et al. (2004)),

 21.2.2 $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)=e^{-\pi[\imagpart{\mathbf{z}}]\cdot[\imagpart{\boldsymbol{{\Omega}}}]% ^{-1}\cdot[\imagpart{\mathbf{z}}]}\mathop{\theta\/}\nolimits\!\left(\mathbf{z}% \middle|\boldsymbol{{\Omega}}\right).$ Defines: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$: scaled Riemann theta function Symbols: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$: Riemann theta function, $e$: base of exponential function, $\imagpart{}$: imaginary part and $\boldsymbol{{\Omega}}$: a Riemann matrix Permalink: http://dlmf.nist.gov/21.2.E2 Encodings: TeX, pMML, png

$\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ 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 $\mathop{\theta\/}\nolimits\!\left(z_{1},z_{2}\middle|\begin{bmatrix}i&-\tfrac{% 1}{2}\\ -\tfrac{1}{2}&i\end{bmatrix}\right)=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=-% \infty}^{\infty}e^{-\pi(n_{1}^{2}+n_{2}^{2})}e^{-i\pi n_{1}n_{2}}e^{2\pi i(n_{% 1}z_{1}+n_{2}z_{2})}.$

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

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

§21.2(ii) Riemann Theta Functions with Characteristics

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

 21.2.5 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{% \beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=% \sum_{\mathbf{n}\in\Integer^{g}}e^{2\pi i\left(\frac{1}{2}[\mathbf{n}+% \boldsymbol{{\alpha}}]\cdot\boldsymbol{{\Omega}}\cdot[\mathbf{n}+\boldsymbol{{% \alpha}}]+[\mathbf{n}+\boldsymbol{{\alpha}}]\cdot[\mathbf{z}+\boldsymbol{{% \beta}}]\right)}.$ Defines: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{% \beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$: Riemann theta function with characteristics Symbols: $\in$: element of, $e$: base of exponential function, $\Integer$: 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

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

and

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

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