# §23.5 Special Lattices

## §23.5(i) Real-Valued Functions

The Weierstrass functions take real values on the real axis iff the lattice is fixed under complex conjugation: $\mathbb{L}=\overline{\mathbb{L}}$; equivalently, when $g_{2},g_{3}\in\mathbb{R}$. This happens in the cases treated in the following four subsections.

## §23.5(ii) Rectangular Lattice

This occurs when both $\omega_{1}$ and $\omega_{3}/i$ are real and positive. Then $\Delta>0$ and the parallelogram with vertices at $0$, $2\omega_{1}$, $2\omega_{1}+2\omega_{3}$, $2\omega_{3}$ is a rectangle.

In this case the lattice roots $e_{1}$, $e_{2}$, and $e_{3}$ are real and distinct. When they are identified as in (23.3.9)

 23.5.1 $\displaystyle e_{1}$ $\displaystyle>e_{2}>e_{3},$ $\displaystyle e_{1}$ $\displaystyle>0>e_{3}.$ ⓘ Symbols: $e_{j}$: zeros Referenced by: (a) Permalink: http://dlmf.nist.gov/23.5.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 23.5(ii), 23.5 and 23

Also, $e_{2}$ and $g_{3}$ have opposite signs unless $\omega_{3}=i\omega_{1}$, in which event both are zero.

As functions of $\Im\omega_{3}$, $e_{1}$ and $e_{2}$ are decreasing and $e_{3}$ is increasing.

## §23.5(iii) Lemniscatic Lattice

This occurs when $\omega_{1}$ is real and positive and $\omega_{3}=i\omega_{1}$. The parallelogram $0$, $2\omega_{1}$, $2\omega_{1}+2\omega_{3}$, $2\omega_{3}$ is a square, and

 23.5.2 $\eta_{1}=i\eta_{3}=\pi/(4\omega_{1}),$
 23.5.3 $\displaystyle e_{1}$ $\displaystyle=-e_{3}=\left(\Gamma\left(\tfrac{1}{4}\right)\right)^{4}/(32\pi% \omega_{1}^{2}),$ $\displaystyle e_{2}$ $\displaystyle=0,$
 23.5.4 $\displaystyle g_{2}$ $\displaystyle=\left(\Gamma\left(\tfrac{1}{4}\right)\right)^{8}/(256\pi^{2}% \omega_{1}^{4}),$ $\displaystyle g_{3}$ $\displaystyle=0.$

Note also that in this case $\tau=\mathrm{i}$. In consequence,

 23.5.5 $\displaystyle k^{2}$ $\displaystyle=\tfrac{1}{2},$ $\displaystyle K\left(k\right)$ $\displaystyle=K'\left(k\right)=\ifrac{\left(\Gamma\left(\tfrac{1}{4}\right)% \right)^{2}}{\left(4\sqrt{\pi}\right)}.$

## §23.5(iv) Rhombic Lattice

This occurs when $\omega_{1}$ is real and positive, $\Im\omega_{3}>0$, $\Re\omega_{3}=\tfrac{1}{2}\omega_{1}$, and $\Delta<0$. The parallelogram $0$, $2\omega_{1}-2\omega_{3}$, $2\omega_{1}$, $2\omega_{3}$, is a rhombus: see Figure 23.5.1.

The lattice root $e_{1}$ is real, and $e_{3}=\bar{e}_{2}$, with $\Im e_{2}>0$. $e_{1}$ and $g_{3}$ have the same sign unless $2\omega_{3}=(1+i)\omega_{1}$ when both are zero: the pseudo-lemniscatic case. As a function of $\Im e_{3}$ the root $e_{1}$ is increasing. For the case $\omega_{3}=e^{\pi i/3}\omega_{1}$ see §23.5(v).

## §23.5(v) Equianharmonic Lattice

This occurs when $\omega_{1}$ is real and positive and $\omega_{3}=e^{\pi i/3}\omega_{1}$. The rhombus $0$, $2\omega_{1}-2\omega_{3}$, $2\omega_{1}$, $2\omega_{3}$ can be regarded as the union of two equilateral triangles: see Figure 23.5.2.

 23.5.6 $\eta_{1}=e^{\pi i/3}\eta_{3}=\frac{\pi}{2\sqrt{3}\omega_{1}},$

and the lattice roots and invariants are given by

 23.5.7 $e_{1}=e^{2\pi i/3}e_{3}=e^{-2\pi i/3}e_{2}=\frac{\left(\Gamma\left(\tfrac{1}{3% }\right)\right)^{6}}{2^{14/3}\pi^{2}\omega_{1}^{2}},$
 23.5.8 $\displaystyle g_{2}$ $\displaystyle=0,$ $\displaystyle g_{3}$ $\displaystyle=\frac{\left(\Gamma\left(\tfrac{1}{3}\right)\right)^{18}}{(4\pi% \omega_{1})^{6}}.$

Note also that in this case $\tau=e^{\mathrm{i}\pi/3}$. In consequence,

 23.5.9 $\displaystyle k^{2}$ $\displaystyle=e^{\mathrm{i}\pi/3},$ $\displaystyle K\left(k\right)$ $\displaystyle=e^{\mathrm{i}\pi/6}K'\left(k\right)=e^{\mathrm{i}\pi/12}\frac{3^% {1/4}\left(\Gamma\left(\frac{1}{3}\right)\right)^{3}}{2^{7/3}\pi}.$