§21.2 Definitions

Permalink:: http://dlmf.nist.gov/21.2

§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

Keywords:: Riemann theta functions, genus, scaled Riemann theta functions, theta functions
Permalink:: http://dlmf.nist.gov/21.2.i

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 $\mathop{\theta\/}\nolimits\!\left(z_{1},z_{2}\middle|\begin{bmatrix}i&-\tfrac{1}{2}\\ -\tfrac{1}{2}&i\end{bmatrix}\right)=\sum _{{n_{1}=-\infty}}^{{\infty}}\sum _{{n_{2}=-\infty}}^{{\infty}}e^{{-\pi(n_{1}^{2}+n_{2}^{2})}}e^{{-i\pi n_{1}n_{2}}}e^{{2\pi i(n_{1}z_{1}+n_{2}z_{2})}}.$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function and $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/21.2.E3
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : scaled Riemann theta function and $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/21.2.E4
Encodings:: TeX, pMML, png

§21.2(ii) Riemann Theta Functions with Characteristics

Keywords:: Riemann theta functions, Riemann theta functions with characteristics, characteristics, theta functions
Referenced by:: §20.11(iv)
Permalink:: http://dlmf.nist.gov/21.2.ii

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

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $e$ : base of exponential function, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.2.E6
Encodings:: TeX, pMML, png

and

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

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.2.E7
Encodings:: TeX, pMML, png

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

Keywords:: Riemann theta functions
Permalink:: http://dlmf.nist.gov/21.2.iii

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

21.2.8 $\mathop{\theta\/}\nolimits\!\left(z\middle|\Omega\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(\pi z\middle|\Omega\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function and $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function
Permalink:: http://dlmf.nist.gov/21.2.E8
Encodings:: TeX, pMML, png

21.2.9 $\mathop{\theta _{{1}}\/}\nolimits\!\left(\pi z\middle|\Omega\right)=-\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}}{\frac{1}{2}}\/}\nolimits\!\left(z\middle|\Omega\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function and $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics
Permalink:: http://dlmf.nist.gov/21.2.E9
Encodings:: TeX, pMML, png

21.2.10 $\mathop{\theta _{{2}}\/}\nolimits\!\left(\pi z\middle|\Omega\right)=\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}}{0}\/}\nolimits\!\left(z\middle|\Omega\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function and $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics
Permalink:: http://dlmf.nist.gov/21.2.E10
Encodings:: TeX, pMML, png

21.2.11 $\mathop{\theta _{{3}}\/}\nolimits\!\left(\pi z\middle|\Omega\right)=\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{0}{0}\/}\nolimits\!\left(z\middle|\Omega\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function and $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics
Permalink:: http://dlmf.nist.gov/21.2.E11
Encodings:: TeX, pMML, png

21.2.12 $\mathop{\theta _{{4}}\/}\nolimits\!\left(\pi z\middle|\Omega\right)=\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{0}{\tfrac{1}{2}}\/}\nolimits\!\left(z\middle|\Omega\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function and $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics
Permalink:: http://dlmf.nist.gov/21.2.E12
Encodings:: TeX, pMML, png

§21.2 Definitions

Contents

§21.2(i) Riemann Theta Functions

¶ Example

§21.2(ii) Riemann Theta Functions with Characteristics

§21.2(iii) Relation to Classical Theta Functions