# §23.3 Differential Equations

## §23.3(i) Invariants, Roots, and Discriminant

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 discriminant1.11(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 $e_{j}=\wp\left(\omega_{j}|\mathbb{L}\right),$ $j=1,2,3$.

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

## §23.3(ii) Differential Equations and Derivatives

 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 ${\wp'}^{2}(z)=4(\wp\left(z\right)-e_{1})(\wp\left(z\right)-e_{2})(\wp\left(z% \right)-e_{3}),$
 23.3.12 $\wp''\left(z\right)=6{\wp}^{2}\left(z\right)-\tfrac{1}{2}g_{2},$
 23.3.13 $\wp'''\left(z\right)=12\wp\left(z\right)\wp'\left(z\right).$