21 Multidimensional Theta Functions Notation 21 Multidimensional Theta Functions 21.2 Definitions

§21.1 Special Notation

ⓘ

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).

21 Multidimensional Theta Functions 21.2 Definitions

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov