§23.2 Definitions and Periodic Properties

Referenced by:: §19.25(vi), ¶ ‣ §31.2(iv), §32.13(iii)
Permalink:: http://dlmf.nist.gov/23.2

§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity

§23.2(i) Lattices

Notes:: See Walker (1996, §3.1).
Keywords:: Weierstrass elliptic functions, lattice
Permalink:: http://dlmf.nist.gov/23.2.i

If $\omega _{1}$ and $\omega _{3}$ are nonzero real or complex numbers such that $\imagpart{(\omega _{3}/\omega _{1})}>0$ , then the set of points $2m\omega _{1}+2n\omega _{3}$ , with $m,n\in\Integer$ , 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
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then $2\omega _{2}$ , $2\omega _{3}$ are generators, as are $2\omega _{2}$ , $2\omega _{1}$ . In general, if

23.2.2

$\chi _{1}=a\omega _{1}+b\omega _{3},$

$\chi _{3}=c\omega _{1}+d\omega _{3},$

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
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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
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§23.2(ii) Weierstrass Elliptic Functions

Notes:: See Whittaker and Watson (1927, §§20.2, 20.4, 20.42), Lawden (1989, Chapter 6).
Keywords:: $\mathop{\wp\/}\nolimits$ -function, Weierstrass elliptic functions, sigma function
Referenced by:: §29.2(ii)
Permalink:: http://dlmf.nist.gov/23.2.ii

23.2.4 $\mathop{\wp\/}\nolimits\!\left(z\right)=\frac{1}{z^{2}}+\sum _{{w\in\mathbb{L}\setminus\{ 0\}}}\left(\frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right),$

Defines:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function
Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\in$ : element of, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
Referenced by:: §23.9, Version 1.0.5 (October 1, 2012)
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

23.2.5 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{1}{z}+\sum _{{w\in\mathbb{L}\setminus\{ 0\}}}\left(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right),$

Defines:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function
Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass zeta function, $\in$ : element of, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.2.E5
Encodings:: TeX, pMML, png

23.2.6 $\mathop{\sigma\/}\nolimits\!\left(z\right)=z\prod _{{w\in\mathbb{L}\setminus\{ 0\}}}\left(\left(1-\frac{z}{w}\right)\mathop{\exp\/}\nolimits\!\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).$

Defines:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function
Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass sigma function, $\in$ : element of, $\mathop{\exp\/}\nolimits z$ : exponential function, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice, $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
Referenced by:: §23.2(iii)
Permalink:: http://dlmf.nist.gov/23.2.E6
Encodings:: TeX, pMML, png

The double series and double product are absolutely and uniformly convergent in compact sets in $\Complex$ 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 $\mathop{\wp\/}\nolimits\!\left(z\right)=-{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(z\right),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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

23.2.8 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\ifrac{{\mathop{\sigma\/}\nolimits^{{\prime}}}\!\left(z\right)}{\mathop{\sigma\/}\nolimits\!\left(z\right)}.$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): 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
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$\mathop{\wp\/}\nolimits\!\left(z\right)$ and $\mathop{\zeta\/}\nolimits\!\left(z\right)$ are meromorphic functions with poles at the lattice points. $\mathop{\wp\/}\nolimits\!\left(z\right)$ is even and $\mathop{\zeta\/}\nolimits\!\left(z\right)$ is odd. The poles of $\mathop{\wp\/}\nolimits\!\left(z\right)$ are double with residue 0; the poles of $\mathop{\zeta\/}\nolimits\!\left(z\right)$ are simple with residue 1. The function $\mathop{\sigma\/}\nolimits\!\left(z\right)$ 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 $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ , $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ , and $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ , respectively.

§23.2(iii) 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.
Keywords:: Weierstrass elliptic functions, elliptic functions
Referenced by:: §21.8, §23.6(iii), §32.10(vi)
Permalink:: http://dlmf.nist.gov/23.2.iii

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 $\mathop{\wp\/}\nolimits\!\left(z+2\omega _{j}\right)=\mathop{\wp\/}\nolimits\!\left(z\right),$ $j=1,2,3$ .

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -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

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

23.2.10 ${\mathop{\wp\/}\nolimits^{{\prime}}}\!\left(\omega _{j}\right)=0,$ $j=1,2,3$ .

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -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

The function $\mathop{\zeta\/}\nolimits\!\left(z\right)$ is quasi-periodic: for $j=1,2,3$ ,

23.2.11 $\mathop{\zeta\/}\nolimits\!\left(z+2\omega _{j}\right)=\mathop{\zeta\/}\nolimits\!\left(z\right)+2\eta _{j},$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
A&S Ref:: 18.2.19
Permalink:: http://dlmf.nist.gov/23.2.E11
Encodings:: TeX, pMML, png

where

23.2.12 $\eta _{j}=\mathop{\zeta\/}\nolimits\!\left(\omega _{j}\right).$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z\right)$ (= $\mathop{\zeta\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\mathbb{L}$ : lattice, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
A&S Ref:: 18.3.3
Permalink:: http://dlmf.nist.gov/23.2.E12
Encodings:: TeX, pMML, png

Also,

23.2.13 $\eta _{1}+\eta _{2}+\eta _{3}=0,$

Symbols:: $\eta _{j}$
Permalink:: http://dlmf.nist.gov/23.2.E13
Encodings:: TeX, pMML, png

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:: $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
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

For $j=1,2,3$ , the function $\mathop{\sigma\/}\nolimits\!\left(z\right)$ satisfies

23.2.15 $\mathop{\sigma\/}\nolimits\!\left(z+2\omega _{j}\right)=-e^{{2\eta _{j}(z+\omega _{j})}}\mathop{\sigma\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $e$ : base of exponential function, $\mathbb{L}$ : lattice, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Referenced by:: §23.2(iii)
Permalink:: http://dlmf.nist.gov/23.2.E15
Encodings:: TeX, pMML, png

23.2.16 ${\mathop{\sigma\/}\nolimits^{{\prime}}}\!\left(2\omega _{j}\right)=-e^{{2\eta _{j}\omega _{j}}}.$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $e$ : base of exponential function, $\mathbb{L}$ : lattice, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $\eta _{j}$
Referenced by:: §23.2(iii)
Permalink:: http://dlmf.nist.gov/23.2.E16
Encodings:: TeX, pMML, png

More generally, if $j=1,2,3$ , $k=1,2,3$ , $j\neq k$ , and $m,n\in\Integer$ , then

23.2.17 $\ifrac{\mathop{\sigma\/}\nolimits\!\left(z+2m\omega _{j}+2n\omega _{k}\right)}{\mathop{\sigma\/}\nolimits\!\left(z\right)}=(-1)^{{m+n+mn}}\mathop{\exp\/}\nolimits\!\left((2m\eta _{j}+2n\eta _{k})(m\omega _{j}+n\omega _{k}+z)\right).$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathbb{L}$ : lattice, $n$ : integer, $m$ : integer, $z$ : complex, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $\eta _{j}$ 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

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

§23.2 Definitions and Periodic Properties

Contents

§23.2(i) Lattices

§23.2(ii) Weierstrass Elliptic Functions

§23.2(iii) Periodicity