Riemann matrix

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1: 21.10 Methods of Computation

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§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

►In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemann surface. …

2: 21.1 Special Notation

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$g,h$	positive integers.
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$𝛀$	$g\times g$ complex, symmetric matrix with $\mathrm{\Im }𝛀$ strictly positive definite, i.e., a Riemann matrix.
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3: 21.4 Graphics

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21.4.1

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Referenced by:

Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1

Permalink:

http://dlmf.nist.gov/21.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

►This Riemann matrix originates from the Riemann surface represented by the algebraic curve ${\mu }^3-{\lambda }^7+2{\lambda }^3\mu =0$; compare §21.7(i). … ►

21.4.2

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

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21.4.3

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve ${\mu }^3+2\mu -{\lambda }^4=0$; compare §21.7(i). Magnify 3D Help

4: 21.3 Symmetry and Quasi-Periodicity

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21.3.1 $$\theta \left(-𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

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21.3.2 $$\theta \left(𝐳+{𝐦}_1\right|𝛀)=\theta \left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

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21.3.3 $$\theta \left(𝐳+{𝐦}_1+𝛀{𝐦}_2\right|𝛀)={\mathrm{e}}^{-2\pi \mathrm{i}\left(\frac{1}{2}{𝐦}_2\cdot 𝛀\cdot {𝐦}_2+{𝐦}_2\cdot 𝐳\right)}\theta \left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

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21.3.4 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶+{𝐦}_1}{𝜷+{𝐦}_2}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}𝜶\cdot {𝐦}_2}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: 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, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Referenced by:

Erratum (V1.0.5) for Equation (21.3.4)

Permalink:

http://dlmf.nist.gov/21.3.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the vector ${𝐦}_2$ on the right-hand-side was given incorrectly as ${𝐦}_1$.

Reported 2012-08-27 by Klaas Vantournhout

See also:

Annotations for §21.3(ii), §21.3 and Ch.21

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21.3.6 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(-𝐳\right|𝛀)={(-1)}^{4𝜶\cdot 𝜷}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(ii), §21.3 and Ch.21

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5: 21.5 Modular Transformations

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21.5.4 $$\theta \left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\xi (𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, Matrix $𝚪$ and $\xi (𝚪)$: eighth root of unity

Referenced by:

§21.5(i)

Permalink:

http://dlmf.nist.gov/21.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

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21.5.5

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $𝛀$: a Riemann matrix and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

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21.5.6

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $g$: positive integer, $𝛀$: a Riemann matrix and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

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21.5.7

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $g$: positive integer, $𝛀$: a Riemann matrix, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Referenced by:

(21.5.8), Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/21.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(i), §21.5 and Ch.21

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21.5.9 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝐃𝜶-𝐂𝜷+\frac{1}{2}\mathrm{diag}[𝐂{𝐃}^{\mathrm{T}}]}{-𝐁𝜶+𝐀𝜷+\frac{1}{2}\mathrm{diag}[𝐀{𝐁}^{\mathrm{T}}]}\right]\left({\left[{[𝐂𝛀+𝐃]}^{-1}\right]}^{\mathrm{T}}𝐳\right|[𝐀𝛀+𝐁]{[𝐂𝛀+𝐃]}^{-1})=\kappa (𝜶,𝜷,𝚪)\sqrt{det[𝐂𝛀+𝐃]}{\mathrm{e}}^{\pi \mathrm{i}𝐳\cdot \left[{[𝐂𝛀+𝐃]}^{-1}𝐂\right]\cdot 𝐳}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $\mathrm{diag}𝐀$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $𝚪$

Permalink:

http://dlmf.nist.gov/21.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.5(ii), §21.5 and Ch.21

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6: 21.9 Integrable Equations

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21.9.4 $$u(x,y,t)=c+2\frac{{\partial }^2}{{\partial x}^2}\ln \left(\theta \left(𝐤x+𝐥y+𝝎t+\mathit{\varphi }\right|𝛀)\right),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝛀$: a Riemann matrix, $u(x,y,t)$: elevation and $c$: complex constant

Referenced by:

§21.9

Permalink:

http://dlmf.nist.gov/21.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.9 and Ch.21

… ►Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). …

7: 21.6 Products

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21.6.3 $$\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}\mathrm{tr}\left[\frac{1}{2}{𝐀}^{\mathrm{T}}𝛀𝐀+{𝐀}^{\mathrm{T}}[𝐙+𝐁]\right]}\prod _{j=1}^h\theta \left({𝐳}_j+𝛀{𝐚}_j+{𝐛}_j\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{tr}𝐀$: trace of matrix, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

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21.6.4 $$\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{\sum }_{k=1}^h{T}_{jk}{𝐜}_k}{{\sum }_{k=1}^h{T}_{jk}{𝐝}_k}\right]\left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{-2\pi \mathrm{i}{\sum }_{j=1}^h{𝐛}_j\cdot {𝐜}_j}\prod _{j=1}^h\theta \left[\genfrac{}{}{0.0pt}{}{{𝐚}_j+{𝐜}_j}{{𝐛}_j+{𝐝}_j}\right]\left({𝐳}_j\right|𝛀),$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

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21.6.6 $$\theta \left(\frac{𝐱+𝐲+𝐮+𝐯}{2}\right|𝛀)\theta \left(\frac{𝐱+𝐲-𝐮-𝐯}{2}\right|𝛀)\theta \left(\frac{𝐱-𝐲+𝐮-𝐯}{2}\right|𝛀)\theta \left(\frac{𝐱-𝐲-𝐮+𝐯}{2}\right|𝛀)=\frac{1}{2^g}\sum _{𝜶\in \frac{1}{2}{\mathbb{Z}}^g/{\mathbb{Z}}^g}\sum _{𝜷\in \frac{1}{2}{\mathbb{Z}}^g/{\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(2𝜶\cdot 𝛀\cdot 𝜶+𝜶\cdot [𝐱+𝐲+𝐮+𝐯]\right)}\theta \left(𝐱+𝛀𝜶+𝜷\right|𝛀)\theta \left(𝐲+𝛀𝜶+𝜷\right|𝛀)\theta \left(𝐮+𝛀𝜶+𝜷\right|𝛀)\theta \left(𝐯+𝛀𝜶+𝜷\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: 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, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

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21.6.7 $$\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[{𝐜}_1+{𝐜}_2+{𝐜}_3+{𝐜}_4]}{\frac{1}{2}[{𝐝}_1+{𝐝}_2+{𝐝}_3+{𝐝}_4]}\right]\left(\frac{𝐱+𝐲+𝐮+𝐯}{2}\right|𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[{𝐜}_1+{𝐜}_2-{𝐜}_3-{𝐜}_4]}{\frac{1}{2}[{𝐝}_1+{𝐝}_2-{𝐝}_3-{𝐝}_4]}\right]\left(\frac{𝐱+𝐲-𝐮-𝐯}{2}\right|𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[{𝐜}_1-{𝐜}_2+{𝐜}_3-{𝐜}_4]}{\frac{1}{2}[{𝐝}_1-{𝐝}_2+{𝐝}_3-{𝐝}_4]}\right]\left(\frac{𝐱-𝐲+𝐮-𝐯}{2}\right|𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[{𝐜}_1-{𝐜}_2-{𝐜}_3+{𝐜}_4]}{\frac{1}{2}[{𝐝}_1-{𝐝}_2-{𝐝}_3+{𝐝}_4]}\right]\left(\frac{𝐱-𝐲-𝐮+𝐯}{2}\right|𝛀)=\frac{1}{2^g}\sum _{𝜶\in \frac{1}{2}{\mathbb{Z}}^g/{\mathbb{Z}}^g}\sum _{𝜷\in \frac{1}{2}{\mathbb{Z}}^g/{\mathbb{Z}}^g}{\mathrm{e}}^{-2\pi \mathrm{i}𝜷\cdot [{𝐜}_1+{𝐜}_2+{𝐜}_3+{𝐜}_4]}\theta \left[\genfrac{}{}{0.0pt}{}{{𝐜}_1+𝜶}{{𝐝}_1+𝜷}\right]\left(𝐱\right|𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{{𝐜}_2+𝜶}{{𝐝}_2+𝜷}\right]\left(𝐲\right|𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{{𝐜}_3+𝜶}{{𝐝}_3+𝜷}\right]\left(𝐮\right|𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{{𝐜}_4+𝜶}{{𝐝}_4+𝜷}\right]\left(𝐯\right|𝛀).$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: 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, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

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21.6.8 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜸}\right]\left({𝐳}_1\right|𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{𝜷}{𝜹}\right]\left({𝐳}_2\right|𝛀)=\sum _{𝝂\in {\mathbb{Z}}^g/\left(2{\mathbb{Z}}^g\right)}\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[𝜶+𝜷+𝝂]}{𝜸+𝜹}\right]\left({𝐳}_1+{𝐳}_2\right|2𝛀)\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}[𝜶-𝜷+𝝂]}{𝜸-𝜹}\right]\left({𝐳}_1-{𝐳}_2\right|2𝛀).$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\in$: element of, $\mathbb{Z}$: set of all integers, $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $𝜹$: $g$-dimensional vector and $𝜸$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(ii), §21.6 and Ch.21

…

8: 21.2 Definitions

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21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

ⓘ

Defines:

$\theta \left(𝐳\right|𝛀)$: 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 $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

pMML, png, TeX

See also:

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

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21.2.2 $$\widehat{\theta }\left(𝐳\right|𝛀)={\mathrm{e}}^{-\pi [\mathrm{\Im }𝐳]\cdot {[\mathrm{\Im }𝛀]}^{-1}\cdot [\mathrm{\Im }𝐳]}\theta \left(𝐳\right|𝛀).$$

ⓘ

Defines:

$\widehat{\theta }\left(𝐳\right|𝛀)$: scaled Riemann theta function

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.2.E2

Encodings:

pMML, png, TeX

See also:

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

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21.2.5 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}[𝐧+𝜶]\cdot 𝛀\cdot [𝐧+𝜶]+[𝐧+𝜶]\cdot [𝐳+𝜷]\right)}.$$

ⓘ

Defines:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: 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, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(ii), §21.2 and Ch.21

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21.2.6 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝜶\cdot 𝛀\cdot 𝜶+𝜶\cdot [𝐳+𝜷]\right)}\theta \left(𝐳+𝛀𝜶+𝜷\right|𝛀),$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: 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, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(ii), §21.2 and Ch.21

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21.2.7 $$\theta \left[\genfrac{}{}{0.0pt}{}{\mathrm{𝟎}}{\mathrm{𝟎}}\right]\left(𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀).$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(ii), §21.2 and Ch.21

…

9: 21.7 Riemann Surfaces

… ►is a Riemann matrix and it is used to define the corresponding Riemann theta function. … ►

21.7.7 $$({\frac{\partial }{\partial z_1}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}},\mathrm{\dots },{\frac{\partial }{\partial z_g}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}})\ne \mathrm{𝟎}.$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.8 $$\zeta ={\sum _{j=1}^g{\omega }_j\frac{\partial }{\partial z_j}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}}.$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector and $\omega$: differential

Permalink:

http://dlmf.nist.gov/21.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.9 $$E(P_1,P_2)=\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left({\int }_{P_1}^{P_2}𝝎\right|𝛀)/\left(\sqrt{\zeta (P_1)}\sqrt{\zeta (P_2)}\right),$$

ⓘ

Defines:

$E(P_1,P_2)$: prime form (locally)

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\int$: integral, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $P_1,P_2$: points and $𝝎$: vector of differentials

Permalink:

http://dlmf.nist.gov/21.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.17 $$\sum _{P_j\in U}\prod _{k=1}^4\theta \left[\genfrac{}{}{0.0pt}{}{{𝜶}_k+{𝜼}^1(P_j)}{{𝜷}_k+{𝜼}^2(P_j)}\right]\left({𝐳}_k\right|𝛀)=\sum _{P_j\in U^c}\prod _{k=1}^4\theta \left[\genfrac{}{}{0.0pt}{}{{𝜶}_k+{𝜼}^1(P_j)}{{𝜷}_k+{𝜼}^2(P_j)}\right]\left({𝐳}_k\right|𝛀).$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\in$: element of, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $P_j$: branch points and $U$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► …