Weierstrass elliptic-function form

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2: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … nome. discriminant ${g_{2}}^{3}-27{g_{3}}^{2}$. …
The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
3: 23.9 Laurent and Other Power Series
§23.9 Laurent and Other Power Series
$c_{2}=\frac{1}{20}g_{2},$
For $j=1,2,3$, and with $e_{j}$ as in §23.3(i),
23.9.6 $\wp\left(\omega_{j}+t\right)=e_{j}+(3e_{j}^{2}-5c_{2})t^{2}+(10c_{2}e_{j}+21c_% {3})t^{4}+(7c_{2}e_{j}^{2}+21c_{3}e_{j}+5c_{2}^{2})t^{6}+O\left(t^{8}\right),$
Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\wp\left(z\right)\to 0$. …
4: 23.21 Physical Applications
The Weierstrass function $\wp$ plays a similar role for cubic potentials in canonical form $g_{3}+g_{2}x-4x^{3}$. …
§23.21(iii) Ellipsoidal Coordinates
where $x,y,z$ are the corresponding Cartesian coordinates and $e_{1}$, $e_{2}$, $e_{3}$ are constants. … Another form is obtained by identifying $e_{1}$, $e_{2}$, $e_{3}$ as lattice roots (§23.3(i)), and setting …
6: 23.3 Differential Equations
The lattice invariants are defined by … and are denoted by $e_{1},e_{2},e_{3}$. … 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. …
7: 23.14 Integrals
§23.14 Integrals
23.14.1 $\int\wp\left(z\right)\mathrm{d}z=-\zeta\left(z\right),$
23.14.2 $\int{\wp}^{2}\left(z\right)\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1}{% 12}g_{2}z,$
23.14.3 $\int{\wp}^{3}\left(z\right)\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-\frac% {3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
8: 23.13 Zeros
§23.13 Zeros
For information on the zeros of $\wp\left(z\right)$ see Eichler and Zagier (1982).
9: 19.25 Relations to Other Functions
§19.25(vi) WeierstrassEllipticFunctions
Let $\mathbb{L}$ be a lattice for the Weierstrass elliptic function $\wp\left(z\right)$. …The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which $\wp\left(z\right)-e_{j}<0$, for some $j$. … for some $2\omega_{j}\in\mathbb{L}$ and $\wp\left(\omega_{j}\right)=e_{j}$. … in which $2\omega_{1}$ and $2\omega_{3}$ are generators for the lattice $\mathbb{L}$, $\omega_{2}=-\omega_{1}-\omega_{3}$, and $\eta_{j}=\zeta\left(\omega_{j}\right)$ (see (23.2.12)). …