23 Weierstrass Elliptic and Modular Functions Weierstrass Elliptic Functions 23.2 Definitions and Periodic Properties 23.4 Graphics

§23.3 Differential Equations

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Referenced by:: §19.25(vi)
Permalink:: http://dlmf.nist.gov/23.3
See also:: Annotations for Ch.23

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

§23.3(i) Invariants, Roots, and Discriminant

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Defines:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\sigma\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass sigma function and $\zeta\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass zeta function
Keywords:: Weierstrass elliptic functions, discriminant, invariants, lattice, roots
Notes:: See Whittaker and Watson (1927, §§20.22, 20.32, 21.73), Lawden (1989, §6.7), Walker (1996, §3.4).
Referenced by:: §19.25(vi), §23.19, §23.21(iii), §23.9, Erratum (V1.0.18) for Subsection 19.25(vi)
Permalink:: http://dlmf.nist.gov/23.3.i
See also:: Annotations for §23.3 and Ch.23

The lattice invariants are defined by

23.3.1	$\displaystyle g_{2}$	$\displaystyle=60\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-4},$
ⓘ Symbols: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\in$ : element of, $\setminus$ : set subtraction and $\mathbb{L}$ : lattice A&S Ref: 18.1.1 Referenced by: §23.3(i) Permalink: http://dlmf.nist.gov/23.3.E1 Encodings: TeX, pMML, png See also: Annotations for §23.3(i), §23.3 and Ch.23
23.3.2	$\displaystyle g_{3}$	$\displaystyle=140\sum_{w\in\mathbb{L}\setminus\{0\}}w^{-6}.$
ⓘ Symbols: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\in$ : element of, $\setminus$ : set subtraction and $\mathbb{L}$ : lattice A&S Ref: 18.1.1 Referenced by: §23.3(i) Permalink: http://dlmf.nist.gov/23.3.E2 Encodings: TeX, pMML, png See also: Annotations for §23.3(i), §23.3 and Ch.23

The lattice roots satisfy the cubic equation

23.3.3		$4z^{3}-g_{2}z-g_{3}=0,$
ⓘ Symbols: $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice 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 See also: Annotations for §23.3(i), §23.3 and Ch.23

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

23.3.4		$\Delta={g_{2}}^{3}-27{g_{3}}^{2}=16(e_{2}-e_{3})^{2}(e_{3}-e_{1})^{2}(e_{1}-e_% {2})^{2}.$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice 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 See also: Annotations for §23.3(i), §23.3 and Ch.23

In consequence,

23.3.5		$e_{1}+e_{2}+e_{3}=0,$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots and $\mathbb{L}$ : lattice A&S Ref: 18.1.11 Referenced by: (23.22.1), §23.22(ii) Permalink: http://dlmf.nist.gov/23.3.E5 Encodings: TeX, pMML, png See also: Annotations for §23.3(i), §23.3 and Ch.23

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: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ and $\mathbb{L}$ : lattice A&S Ref: 18.1.9 Referenced by: §23.22(ii) Permalink: http://dlmf.nist.gov/23.3.E6 Encodings: TeX, pMML, png See also: Annotations for §23.3(i), §23.3 and Ch.23

23.3.7		$g_{3}=4e_{1}e_{2}e_{3}=\tfrac{4}{3}({e_{1}}^{3}+{e_{2}}^{3}+{e_{3}}^{3}).$
ⓘ Symbols: $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ and $\mathbb{L}$ : lattice A&S Ref: 18.1.10 Referenced by: (23.22.1), §23.22(ii), §23.22(ii) Permalink: http://dlmf.nist.gov/23.3.E7 Encodings: TeX, pMML, png See also: Annotations for §23.3(i), §23.3 and Ch.23

Let ${g_{2}}^{3}\neq 27{g_{3}}^{2}$ , 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		$\wp\left(z;g_{2},g_{3}\right)=\wp\left(z\|\mathbb{L}\right).$
ⓘ Defines: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice and $z$ : complex Permalink: http://dlmf.nist.gov/23.3.E8 Encodings: TeX, pMML, png See also: Annotations for §23.3(i), §23.3 and Ch.23

Similarly for $\zeta\left(z;g_{2},g_{3}\right)$ and $\sigma\left(z;g_{2},g_{3}\right)$ . As functions of $g_{2}$ and $g_{3}$ , $\wp\left(z;g_{2},g_{3}\right)$ and $\zeta\left(z;g_{2},g_{3}\right)$ are meromorphic and $\sigma\left(z;g_{2},g_{3}\right)$ 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		$e_{j}=\wp\left(\omega_{j}\|\mathbb{L}\right),$
$j=1,2,3$ .
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice 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 See also: Annotations for §23.3(i), §23.3 and Ch.23

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

§23.3(ii) Differential Equations and Derivatives

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Keywords:: Weierstrass elliptic functions, derivatives, differential equations
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).
Referenced by:: §23.14, §23.9
Permalink:: http://dlmf.nist.gov/23.3.ii
See also:: Annotations for §23.3 and Ch.23

23.3.10		${\wp'}^{2}(z)=4{\wp}^{3}\left(z\right)-g_{2}\wp\left(z\right)-g_{3},$
ⓘ Symbols: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice 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 See also: Annotations for §23.3(ii), §23.3 and Ch.23

23.3.11		${\wp'}^{2}(z)=4(\wp\left(z\right)-e_{1})(\wp\left(z\right)-e_{2})(\wp\left(z% \right)-e_{3}),$
ⓘ Symbols: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice 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 See also: Annotations for §23.3(ii), §23.3 and Ch.23

23.3.12		$\wp''\left(z\right)=6{\wp}^{2}\left(z\right)-\tfrac{1}{2}g_{2},$
ⓘ Symbols: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice 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 See also: Annotations for §23.3(ii), §23.3 and Ch.23

23.3.13		$\wp'''\left(z\right)=12\wp\left(z\right)\wp'\left(z\right).$
ⓘ Symbols: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function, $g_{\NVar{j}}$ : Weierstrass lattice invariants $g_{2}$ , $g_{3}$ , $\mathbb{L}$ : lattice 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 See also: Annotations for §23.3(ii), §23.3 and Ch.23

§23.3 Differential Equations

Contents

§23.3(i) Invariants, Roots, and Discriminant

§23.3(ii) Differential Equations and Derivatives