§23.3 Differential Equations

Permalink:: http://dlmf.nist.gov/23.3

§23.3(i) Invariants, Roots, and Discriminant
§23.3(ii) Differential Equations and Derivatives

§23.3(i) Invariants, Roots, and Discriminant

Notes:: See Whittaker and Watson (1927, §§20.22, 20.32, 21.73), Lawden (1989, §6.7), Walker (1996, §3.4).
Defines:: $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass zeta function and $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass sigma function
Keywords:: Weierstrass elliptic functions, discriminant, lattice, roots
Referenced by:: §23.19, §23.21(iii), §23.9
Permalink:: http://dlmf.nist.gov/23.3.i

The lattice invariants are defined by

23.3.1 $g_{2}=60\sum _{{w\in\mathbb{L}\setminus\{ 0\}}}w^{{-4}},$

Symbols:: $\in$ : element of, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice and $g_{2}$ , $g_{3}$ : lattice invariants
A&S Ref:: 18.1.1
Referenced by:: §23.3(i)
Permalink:: http://dlmf.nist.gov/23.3.E1
Encodings:: TeX, pMML, png

23.3.2 $g_{3}=140\sum _{{w\in\mathbb{L}\setminus\{ 0\}}}w^{{-6}}.$

Symbols:: $\in$ : element of, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\setminus$ : set subtraction, $\mathbb{L}$ : lattice and $g_{2}$ , $g_{3}$ : lattice invariants
A&S Ref:: 18.1.1
Referenced by:: §23.3(i)
Permalink:: http://dlmf.nist.gov/23.3.E2
Encodings:: TeX, pMML, png

The lattice roots satisfy the cubic equation

23.3.3 $4z^{3}-g_{2}z-g_{3}=0,$

Symbols:: $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
A&S Ref:: 18.1.13
Referenced by:: §23.3(ii)
Permalink:: http://dlmf.nist.gov/23.3.E3
Encodings:: TeX, pMML, png

and are denoted by $e_{1},e_{2},e_{3}$ . The discriminant (§1.11(ii)) is given by

23.3.4 $\Delta=g_{2}^{3}-27g_{3}^{2}=16(e_{2}-e_{3})^{2}(e_{3}-e_{1})^{2}(e_{1}-e_{2})^{2}.$

Symbols:: $g_{2}$ , $g_{3}$ : lattice invariants, $e_{j}$ : zeros and $\Delta$ : discriminant
A&S Ref:: 18.1.8
Referenced by:: §23.19
Permalink:: http://dlmf.nist.gov/23.3.E4
Encodings:: TeX, pMML, png

In consequence,

23.3.5 $e_{1}+e_{2}+e_{3}=0,$

Symbols:: $e_{j}$ : zeros
A&S Ref:: 18.1.11
Permalink:: http://dlmf.nist.gov/23.3.E5
Encodings:: TeX, pMML, png

23.3.6 $g_{2}=2(e_{1}^{2}+e_{2}^{2}+e_{3}^{2})=-4(e_{2}e_{3}+e_{3}e_{1}+e_{1}e_{2}),$

Symbols:: $g_{2}$ , $g_{3}$ : lattice invariants and $e_{j}$ : zeros
A&S Ref:: 18.1.9
Referenced by:: ¶ ‣ §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.3.E6
Encodings:: TeX, pMML, png

23.3.7 $g_{3}=4e_{1}e_{2}e_{3}=\tfrac{4}{3}(e_{1}^{3}+e_{2}^{3}+e_{3}^{3}).$

Symbols:: $g_{2}$ , $g_{3}$ : lattice invariants and $e_{j}$ : zeros
A&S Ref:: 18.1.10
Referenced by:: ¶ ‣ §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.3.E7
Encodings:: TeX, pMML, png

Let $g_{2}^{3}\neq 27g_{3}^{2}$ , or equivalently $\Delta$ be nonzero, or $e_{1},e_{2},e_{3}$ be distinct. Given $g_{2}$ and $g_{3}$ there is a unique lattice $\mathbb{L}$ such that (23.3.1) and (23.3.2) are satisfied. We may therefore define

23.3.8 $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)=\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right).$

Defines:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function
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, $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
Permalink:: http://dlmf.nist.gov/23.3.E8
Encodings:: TeX, pMML, png

Similarly for $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ and $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ . As functions of $g_{2}$ and $g_{3}$ , $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ and $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ are meromorphic and $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ is entire.

Conversely, $g_{2}$ , $g_{3}$ , and the set $\{ e_{1},e_{2},e_{3}\}$ are determined uniquely by the lattice $\mathbb{L}$ independently of the choice of generators. However, given any pair of generators $2\omega _{1}$ , $2\omega _{3}$ of $\mathbb{L}$ , and with $\omega _{2}$ defined by (23.2.1), we can identify the $e_{j}$ individually, via

23.3.9 $e_{j}=\mathop{\wp\/}\nolimits\!\left(\omega _{j}|\mathbb{L}\right),$ $j=1,2,3$ .

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, $e_{j}$ : zeros and $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators
A&S Ref:: 18.3.1
Referenced by:: §23.3(i), §23.5(ii), §23.6(i), §23.6(ii), §23.9
Permalink:: http://dlmf.nist.gov/23.3.E9
Encodings:: TeX, pMML, png

In what follows, it will be assumed that (23.3.9) always applies.

§23.3(ii) Differential Equations and Derivatives

Notes:: See Whittaker and Watson (1927, §20.22), Lawden (1989, §6.7), Walker (1996, §3.4). (23.3.11) follows from (23.3.3), (23.3.10). For (23.3.12), (23.3.13) differentiate (23.3.10).
Keywords:: Weierstrass elliptic functions, derivatives, differential equations
Referenced by:: §23.14, §23.9
Permalink:: http://dlmf.nist.gov/23.3.ii

23.3.10 ${{\mathop{\wp\/}\nolimits^{{\prime}}}}^{2}(z)=4\!{\mathop{\wp\/}\nolimits^{{3}}}\!\left(z\right)-g_{2}\mathop{\wp\/}\nolimits\!\left(z\right)-g_{3},$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
A&S Ref:: 18.1.6 and 18.6.3
Referenced by:: ¶ ‣ §23.22(ii), §23.3(ii)
Permalink:: http://dlmf.nist.gov/23.3.E10
Encodings:: TeX, pMML, png

23.3.11 ${{\mathop{\wp\/}\nolimits^{{\prime}}}}^{2}(z)=4(\mathop{\wp\/}\nolimits\!\left(z\right)-e_{1})(\mathop{\wp\/}\nolimits\!\left(z\right)-e_{2})(\mathop{\wp\/}\nolimits\!\left(z\right)-e_{3}),$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $g_{2}$ , $g_{3}$ : lattice invariants, $e_{j}$ : zeros and $z$ : complex
A&S Ref:: 18.1.7
Referenced by:: §23.3(ii)
Permalink:: http://dlmf.nist.gov/23.3.E11
Encodings:: TeX, pMML, png

23.3.12 ${\mathop{\wp\/}\nolimits^{{\prime\prime}}}\!\left(z\right)=6\!{\mathop{\wp\/}\nolimits^{{2}}}\!\left(z\right)-\tfrac{1}{2}g_{2},$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
A&S Ref:: 18.6.4
Referenced by:: §23.3(ii)
Permalink:: http://dlmf.nist.gov/23.3.E12
Encodings:: TeX, pMML, png

23.3.13 ${\mathop{\wp\/}\nolimits^{{\prime\prime\prime}}}\!\left(z\right)=12\mathop{\wp\/}\nolimits\!\left(z\right){\mathop{\wp\/}\nolimits^{{\prime}}}\!\left(z\right).$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $g_{2}$ , $g_{3}$ : lattice invariants and $z$ : complex
A&S Ref:: 18.6.5
Referenced by:: §23.3(ii)
Permalink:: http://dlmf.nist.gov/23.3.E13
Encodings:: TeX, pMML, png

§23.3 Differential Equations

Contents

§23.3(i) Invariants, Roots, and Discriminant

§23.3(ii) Differential Equations and Derivatives