# §23.2 Definitions and Periodic Properties

## §23.2(i) Lattices

If $\omega_{1}$ and $\omega_{3}$ are nonzero real or complex numbers such that $\Im\left(\omega_{3}/\omega_{1}\right)>0$, then the set of points $2m\omega_{1}+2n\omega_{3}$, with $m,n\in\mathbb{Z}$, constitutes a lattice $\mathbb{L}$ with $2\omega_{1}$ and $2\omega_{3}$ lattice generators.

The generators of a given lattice $\mathbb{L}$ are not unique. For example, if

 23.2.1 $\omega_{1}+\omega_{2}+\omega_{3}=0,$ ⓘ Symbols: $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators Referenced by: §23.2(iii), §23.3(i) Permalink: http://dlmf.nist.gov/23.2.E1 Encodings: TeX, pMML, png See also: Annotations for §23.2(i), §23.2 and Ch.23

then $2\omega_{2}$, $2\omega_{3}$ are generators, as are $2\omega_{2}$, $2\omega_{1}$. In general, if

 23.2.2 $\displaystyle\chi_{1}$ $\displaystyle=a\omega_{1}+b\omega_{3},$ $\displaystyle\chi_{3}$ $\displaystyle=c\omega_{1}+d\omega_{3},$

where $a,b,c,d$ are integers, then $2\chi_{1}$, $2\chi_{3}$ are generators of $\mathbb{L}$ iff

 23.2.3 $ad-bc=1.$ ⓘ Symbols: $a$: integer, $b$: integer, $c$: integer and $d$: integer Permalink: http://dlmf.nist.gov/23.2.E3 Encodings: TeX, pMML, png See also: Annotations for §23.2(i), §23.2 and Ch.23

## §23.2(ii) Weierstrass Elliptic Functions

 23.2.4 ${}\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right),$ ⓘ Defines: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$): Weierstrass $\wp$-function Symbols: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$: Weierstrass $\wp$-function, $g_{\NVar{j}}$: Weierstrass lattice invariants $g_{2}$, $g_{3}$, $\in$: element of, $\setminus$: set subtraction, $\mathbb{L}$: lattice and $z$: complex Keywords: Weierstrass $\wp$-function, modular functions Referenced by: §23.9, Erratum (V1.0.5) for Equation (23.2.4) Permalink: http://dlmf.nist.gov/23.2.E4 Encodings: TeX, pMML, png Errata (effective with 1.0.5): Originally the denominator $(z-w)^{2}$ was given incorrectly as $(z-w^{2})$. Reported 2012-02-16 by James D. Walker See also: Annotations for §23.2(ii), §23.2 and Ch.23
 23.2.5 ${}\zeta\left(z\right)=\frac{1}{z}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right),$ ⓘ Defines: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$): Weierstrass zeta function Symbols: $g_{\NVar{j}}$: Weierstrass lattice invariants $g_{2}$, $g_{3}$, $\zeta\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$: Weierstrass zeta function, $\in$: element of, $\setminus$: set subtraction, $\mathbb{L}$: lattice and $z$: complex Keywords: Weierstrass zeta function, modular functions Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.2.E5 Encodings: TeX, pMML, png See also: Annotations for §23.2(ii), §23.2 and Ch.23
 23.2.6 ${}\sigma\left(z\right)=z\prod_{w\in\mathbb{L}\setminus\{0\}}\left(\left(1-% \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).$ ⓘ Defines: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$): Weierstrass sigma function Symbols: $g_{\NVar{j}}$: Weierstrass lattice invariants $g_{2}$, $g_{3}$, $\sigma\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$: Weierstrass sigma function, $\in$: element of, $\exp\NVar{z}$: exponential function, $\setminus$: set subtraction, $\mathbb{L}$: lattice and $z$: complex Keywords: Weierstrass sigma function, modular functions Referenced by: §23.2(iii) Permalink: http://dlmf.nist.gov/23.2.E6 Encodings: TeX, pMML, png See also: Annotations for §23.2(ii), §23.2 and Ch.23

The double series and double product are absolutely and uniformly convergent in compact sets in $\mathbb{C}$ that do not include lattice points. Hence the order of the terms or factors is immaterial.

When $z\notin\mathbb{L}$ the functions are related by

 23.2.7 $\wp\left(z\right)=-\zeta'\left(z\right),$
 23.2.8 $\zeta\left(z\right)=\ifrac{\sigma'\left(z\right)}{\sigma\left(z\right)}.$

$\wp\left(z\right)$ and $\zeta\left(z\right)$ are meromorphic functions with poles at the lattice points. $\wp\left(z\right)$ is even and $\zeta\left(z\right)$ is odd. The poles of $\wp\left(z\right)$ are double with residue $0$; the poles of $\zeta\left(z\right)$ are simple with residue $1$. The function $\sigma\left(z\right)$ is entire and odd, with simple zeros at the lattice points. When it is important to display the lattice with the functions they are denoted by $\wp\left(z|\mathbb{L}\right)$, $\zeta\left(z|\mathbb{L}\right)$, and $\sigma\left(z|\mathbb{L}\right)$, respectively.

## §23.2(iii) Periodicity

If $2\omega_{1}$, $2\omega_{3}$ is any pair of generators of $\mathbb{L}$, and $\omega_{2}$ is defined by (23.2.1), then

 23.2.9 $\wp\left(z+2\omega_{j}\right)=\wp\left(z\right),$ $j=1,2,3$.

Hence $\wp\left(z\right)$ is an elliptic function, that is, $\wp\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\wp\left(z\right)$ is meromorphic and has two periods whose ratio is not real. We also have

 23.2.10 $\wp'\left(\omega_{j}\right)=0,$ $j=1,2,3$.

The function $\zeta\left(z\right)$ is quasi-periodic: for $j=1,2,3$,

 23.2.11 $\zeta\left(z+2\omega_{j}\right)=\zeta\left(z\right)+2\eta_{j},$

where

 23.2.12 $\eta_{j}=\zeta\left(\omega_{j}\right).$

Also,

 23.2.13 $\eta_{1}+\eta_{2}+\eta_{3}=0,$ ⓘ Symbols: $\eta_{j}$: complex number Permalink: http://dlmf.nist.gov/23.2.E13 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23
 23.2.14 $\eta_{3}\omega_{2}-\eta_{2}\omega_{3}=\eta_{2}\omega_{1}-\eta_{1}\omega_{2}=% \eta_{1}\omega_{3}-\eta_{3}\omega_{1}=\tfrac{1}{2}\pi i.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\omega_{1}$, $\omega_{3}$, $\omega_{2}=-\omega_{1}-\omega_{3}$: lattice generators and $\eta_{j}$: complex number A&S Ref: 18.3.37 Referenced by: §23.12, §23.8(ii) Permalink: http://dlmf.nist.gov/23.2.E14 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

For $j=1,2,3$, the function $\sigma\left(z\right)$ satisfies

 23.2.15 $\sigma\left(z+2\omega_{j}\right)=-e^{2\eta_{j}(z+\omega_{j})}\sigma\left(z% \right),$
 23.2.16 $\sigma'\left(2\omega_{j}\right)=-e^{2\eta_{j}\omega_{j}}.$

More generally, if $j=1,2,3$, $k=1,2,3$, $j\neq k$, and $m,n\in\mathbb{Z}$, then

 23.2.17 $\ifrac{\sigma\left(z+2m\omega_{j}+2n\omega_{k}\right)}{\sigma\left(z\right)}=(% -1)^{m+n+mn}\exp\left((2m\eta_{j}+2n\eta_{k})(m\omega_{j}+n\omega_{k}+z)\right).$

For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).