21 Multidimensional Theta Functions Properties 21.1 Special Notation 21.3 Symmetry and Quasi-Periodicity

§21.2 Definitions

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Permalink:: http://dlmf.nist.gov/21.2
See also:: Annotations for Ch.21

§21.2(i) Riemann Theta Functions
§21.2(ii) Riemann Theta Functions with Characteristics
§21.2(iii) Relation to Classical Theta Functions

§21.2(i) Riemann Theta Functions

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Keywords:: Riemann, Riemann theta functions, analytic properties, components, definition, dimension, genus, scaled, scaled Riemann theta functions, theta functions
Referenced by:: Erratum (V1.0.5) for Subsection 21.2(i), Erratum (V1.0.5) for Clarifications
Permalink:: http://dlmf.nist.gov/21.2.i
Addition (effective with 1.0.5):: The cross-reference to 21.10(i) was added at the end of the second paragraph in this subsection.
See also:: Annotations for §21.2 and Ch.21

21.2.1 $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in% {\mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}% }\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}.$

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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, : 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 -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 components, a -dimensional theta function or as a genus 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 $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{-\pi[\Im% \mathbf{z}]\cdot[\Im\boldsymbol{{\Omega}}]^{-1}\cdot[\Im\mathbf{z}]}\theta% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

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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

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See also:: Annotations for §21.2(i), §21.2 and Ch.21

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

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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 and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/21.2.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §21.2(i), §21.2(i), §21.2 and Ch.21

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

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

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Symbols:: $\hat{\theta}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : scaled Riemann theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/21.2.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §21.2(i), §21.2(i), §21.2 and Ch.21

§21.2(ii) Riemann Theta Functions with Characteristics

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Keywords:: Riemann theta functions, Riemann theta functions with characteristics, Riemann with characteristics, characteristics, half-period, theta functions
Referenced by:: §20.11(iv)
Permalink:: http://dlmf.nist.gov/21.2.ii
See also:: Annotations for §21.2 and Ch.21

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

21.2.5 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in{% \mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}[\mathbf{n}+\boldsymbol{{\alpha}}]% \cdot\boldsymbol{{\Omega}}\cdot[\mathbf{n}+\boldsymbol{{\alpha}}]+[\mathbf{n}+% \boldsymbol{{\alpha}}]\cdot[\mathbf{z}+\boldsymbol{{\beta}}]\right)}.$

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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, : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : -dimensional vector and $\boldsymbol{{\beta}}$ : -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 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\frac{1}{2% }\boldsymbol{{\alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+% \boldsymbol{{\alpha}}\cdot[\mathbf{z}+\boldsymbol{{\beta}}]\right)}\theta\left% (\mathbf{z}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}% \middle|\boldsymbol{{\Omega}}\right),$

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Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : -dimensional vector and $\boldsymbol{{\beta}}$ : -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 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{0}}}{\boldsymbol{{0}}}\left(\mathbf% {z}\middle|\boldsymbol{{\Omega}}\right)=\theta\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right).$

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Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics and $\boldsymbol{{\Omega}}$ : 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 $\tfrac{1}{2}$ are called half-period characteristics. For given $\boldsymbol{{\Omega}}$ , there are $2^{2g}$ -dimensional Riemann theta functions with half-period characteristics.