rectangular case

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7 matching pages

1: 23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of

\wp\left(z\right)

\wp'\left(z\right)

, and

\zeta\left(z\right)

to 7 or 8D in the rectangular and rhombic cases, normalized so that

\omega_{1}=1

and

\omega_{3}=ia

(rectangular case), or

\omega_{1}=1

and

\omega_{3}=\tfrac{1}{2}+ia

(rhombic case), for

a

= 1. …

2: 20.1 Special Notation

… ► ►►►►

$m$ , $n$	integers.
…
$q$ $(\in\mathbb{C})$	the nome, $q=e^{i\pi\tau}$ , $0<\left\|q\right\|<1$ . Since $\tau$ is not a single-valued function of $q$ , it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\Re\tau=0$ , $\Im\tau>0$ , so that $0<q<1$ and $\tau$ and $q$ are uniquely related.
$q^{\alpha}$	$e^{i\alpha\pi\tau}$ for $\alpha\in\mathbb{R}$ (resolving issues of choice of branch).
…

…

3: 23.7 Quarter Periods

… ►where

k,k^{\prime}

and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.

4: 20.2 Definitions and Periodic Properties

… ►

…

Figure 20.2.1:

z

-plane. …Left-hand diagram is the rectangular case (

\tau

purely imaginary); right-hand diagram is the general case. …

…

5: 23.20 Mathematical Applications

… ►These cases correspond to rhombic and rectangular lattices, respectively. …

6: 23.5 Special Lattices

… ►This happens in the cases treated in the following four subsections. ►

§23.5(ii) Rectangular Lattice

… ►

§23.5(iii) Lemniscatic Lattice

… ►

§23.5(iv) Rhombic Lattice

… ►

§23.5(v) Equianharmonic Lattice

…

7: 4.15 Graphics

… ►Lines parallel to the real axis in the

z

-plane map onto ellipses in the

w

-plane with foci at

w=\pm 1

, and lines parallel to the imaginary axis in the

z

-plane map onto rectangular hyperbolas confocal with the ellipses. … ►they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by

-\tfrac{1}{2}\pi

parallel to the

x

-axis, and adjusting the phase coloring in the case of Figure 4.15.10. …