21 Multidimensional Theta Functions Notation 21 Multidimensional Theta Functions 21.2 Definitions

§21.1 Special Notation

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Defines:: $/$ : $S_{1}/S_{2}$ : set of all elements of $S_{1}$ modulo elements of $S_{2}$
Keywords:: Riemann, Riemann matrix, Riemann surface, Riemann theta functions, Riemann theta functions with characteristics, intersection indices, matrix, multidimensional theta functions, notation
Permalink:: http://dlmf.nist.gov/21.1
See also:: Annotations for Ch.21

(For other notation see Notation for the Special Functions.)

$g, h$	positive integers.
${\mathbb{Z}}^{g}$	$\mathbb{Z}\times\mathbb{Z}\times\cdots\times\mathbb{Z}$ ( $g$ times).
${\mathbb{R}}^{g}$	$\mathbb{R}\times\mathbb{R}\times\cdots\times\mathbb{R}$ ( $g$ times).
${\mathbb{Z}}^{g\times h}$	set of all $g\times h$ matrices with integer elements.
$\boldsymbol{{\Omega}}$	$g\times g$ complex, symmetric matrix with $\Im\boldsymbol{{\Omega}}$ strictly positive definite, i.e., a Riemann matrix.
$\boldsymbol{{\alpha}},\boldsymbol{{\beta}}$	$g$ -dimensional vectors, with all elements in $[0,1)$ , unless stated otherwise.
$a_{j}$	$j$ th element of vector $\mathbf{a}$ .
$A_{jk}$	$(j,k)$ th element of matrix $\mathbf{A}$ .
$\mathbf{a}\cdot\mathbf{b}$	scalar product of the vectors $\mathbf{a}$ and $\mathbf{b}$ .
$\mathbf{a}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{b}$	$[\boldsymbol{{\Omega}}\mathbf{a}]\cdot\mathbf{b}=[\boldsymbol{{\Omega}}\mathbf% {b}]\cdot\mathbf{a}$ .
$\operatorname{diag}\mathbf{A}$	Transpose of $[A_{11},A_{22},\dots,A_{gg}]$ .
$\boldsymbol{{0}}_{g}$	$g\times g$ zero matrix.
$\mathbf{I}_{g}$	$g\times g$ identity matrix.
$\mathbf{J}_{2g}$	$\begin{bmatrix}\boldsymbol{{0}}_{g}&\mathbf{I}_{g}\\ -\mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$ .
$S^{g}$	set of $g$ -dimensional vectors with elements in $S$ .
$\|S\|$	number of elements of the set $S$ .
$S_{1}S_{2}$	set of all elements of the form “ $\mbox{element of $S_{1}$}\times\mbox{element of $S_{2}$}$ ”.
$S_{1}/S_{2}$	set of all elements of $S_{1}$ , modulo elements of $S_{2}$ . Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$ . (For an example see §20.12(ii).)
$a\circ b$	intersection index of $a$ and $b$ , two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$ .
$\oint_{a}\omega$	line integral of the differential $\omega$ over the cycle $a$ .

Lowercase boldface letters or numbers are $g$ -dimensional real or complex vectors, either row or column depending on the context. Uppercase boldface letters are $g\times g$ real or complex matrices.

The main functions treated in this chapter are the Riemann theta functions $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ , and the Riemann theta functions with characteristics $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ .

The function $\Theta(\boldsymbol{{\phi}}|\mathbf{B})=\theta\left(\boldsymbol{{\phi}}/(2\pi i% )\middle|\mathbf{B}/(2\pi i)\right)$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).