23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.1 Special Notation 23.3 Differential Equations

§23.2 Definitions and Periodic Properties

ⓘ

Referenced by:: §19.25(vi), §31.2(iv), §32.13(iii)
Permalink:: http://dlmf.nist.gov/23.2
See also:: Annotations for Ch.23

§23.2(i) Lattices

ⓘ

Keywords:: Weierstrass elliptic functions, for elliptic functions, generators, lattice, points
Notes:: See Walker (1996, §3.1).
Referenced by:: §19.25(vi), Erratum (V1.0.18) for Subsection 19.25(vi)
Permalink:: http://dlmf.nist.gov/23.2.i
See also:: Annotations for §23.2 and Ch.23

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	$\displaystyle\chi_{1}$	$\displaystyle=a\omega_{1}+b\omega_{3},$
23.2.2	$\displaystyle\chi_{3}$	$\displaystyle=c\omega_{1}+d\omega_{3},$
ⓘ Symbols: $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $\chi_{j}$ : lattice generators, $a$ : integer, $b$ : integer, $c$ : integer and $d$ : integer Permalink: http://dlmf.nist.gov/23.2.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.2(i), §23.2 and Ch.23

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

ⓘ

Keywords:: $\wp$ -function, Weierstrass elliptic functions, analytic properties, definitions, poles, sigma function, zeros
Notes:: See Whittaker and Watson (1927, §§20.2, 20.4, 20.42), Lawden (1989, Chapter 6).
Referenced by:: §19.25(vi), §29.2(ii)
Permalink:: http://dlmf.nist.gov/23.2.ii
See also:: Annotations for §23.2 and Ch.23

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		$\wp\left(z\right)=-\zeta'\left(z\right),$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice and $z$ : complex A&S Ref: 18.6.1 Referenced by: §23.14, §23.3(ii) Permalink: http://dlmf.nist.gov/23.2.E7 Encodings: TeX, pMML, png See also: Annotations for §23.2(ii), §23.2 and Ch.23

23.2.8		$\zeta\left(z\right)=\ifrac{\sigma'\left(z\right)}{\sigma\left(z\right)}.$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice and $z$ : complex A&S Ref: 18.6.2 Referenced by: §23.3(ii), §23.9 Permalink: http://dlmf.nist.gov/23.2.E8 Encodings: TeX, pMML, png See also: Annotations for §23.2(ii), §23.2 and Ch.23

$\wp\left(z\right)$ 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

ⓘ

Keywords:: Weierstrass elliptic functions, elliptic functions, general, periodicity, quasi-periodicity
Notes:: See Whittaker and Watson (1927, §§20.21, 20.41–20.421), Lawden (1989, §§6.2 and 6.6). For (23.2.16) differentiate (23.2.15) and use (23.2.6). For (23.2.17) use (23.2.15) and induction.
Referenced by:: §21.8, §23.6(iii), §32.10(vi)
Permalink:: http://dlmf.nist.gov/23.2.iii
See also:: Annotations for §23.2 and Ch.23

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		$\wp\left(z+2\omega_{j}\right)=\wp\left(z\right),$
$j=1,2,3$ .
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\mathbb{L}$ : lattice, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.2.18 Permalink: http://dlmf.nist.gov/23.2.E9 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

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		$\wp'\left(\omega_{j}\right)=0,$
$j=1,2,3$ .
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\mathbb{L}$ : lattice and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators A&S Ref: 18.3.2 Referenced by: §23.9 Permalink: http://dlmf.nist.gov/23.2.E10 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

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

23.2.11		$\zeta\left(z+2\omega_{j}\right)=\zeta\left(z\right)+2\eta_{j},$
ⓘ Symbols: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number A&S Ref: 18.2.19 Permalink: http://dlmf.nist.gov/23.2.E11 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

where

23.2.12		$\eta_{j}=\zeta\left(\omega_{j}\right).$
ⓘ Symbols: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z\|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number A&S Ref: 18.3.3 Referenced by: §19.25(vi) Permalink: http://dlmf.nist.gov/23.2.E12 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

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		$\sigma\left(z+2\omega_{j}\right)=-e^{2\eta_{j}(z+\omega_{j})}\sigma\left(z% \right),$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathrm{e}$ : base of natural logarithm, $\mathbb{L}$ : lattice, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Referenced by: §23.2(iii) Permalink: http://dlmf.nist.gov/23.2.E15 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

23.2.16		$\sigma'\left(2\omega_{j}\right)=-e^{2\eta_{j}\omega_{j}}.$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathrm{e}$ : base of natural logarithm, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number Referenced by: §23.2(iii) Permalink: http://dlmf.nist.gov/23.2.E16 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

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

23.2.17		$\ifrac{\sigma\left(z+2m\omega_{j}+2n\omega_{k}\right)}{\sigma\left(z\right)}=(% -1)^{m+n+mn}\exp\left((2m\eta_{j}+2n\eta_{k})(m\omega_{j}+n\omega_{k}+z)\right).$
ⓘ Symbols: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z\|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\exp\NVar{z}$ : exponential function, $\mathbb{L}$ : lattice, $n$ : integer, $m$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $\eta_{j}$ : complex number and $k$ : modulus A&S Ref: 18.2.20, 18.2.21 Referenced by: §23.2(iii) Permalink: http://dlmf.nist.gov/23.2.E17 Encodings: TeX, pMML, png See also: Annotations for §23.2(iii), §23.2 and Ch.23

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

23.1 Special Notation 23.3 Differential Equations

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

§23.2 Definitions and Periodic Properties

Contents

§23.2(i) Lattices

§23.2(ii) Weierstrass Elliptic Functions

§23.2(iii) Periodicity

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!