# invariants

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##### 1: 23.3 Differential Equations
###### §23.3(i) Invariants, Roots, and Discriminant
The lattice invariants are defined by … Given $g_{2}$ and $g_{3}$ there is a unique lattice $\mathbb{L}$ such that (23.3.1) and (23.3.2) are satisfied. …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. …
##### 2: 23.4 Graphics Figure 23.4.1: ℘ ⁡ ( x ; g 2 ⁡ , 0 ) for 0 ≤ x ≤ 9 , g 2 ⁡ = 0. … Magnify Figure 23.4.2: ℘ ⁡ ( x ; 0 , g 3 ⁡ ) for 0 ≤ x ≤ 9 , g 3 ⁡ = 0. … Magnify Figure 23.4.3: ζ ⁡ ( x ; g 2 ⁡ , 0 ) for 0 ≤ x ≤ 8 , g 2 ⁡ = 0. … Magnify Figure 23.4.4: ζ ⁡ ( x ; 0 , g 3 ⁡ ) for 0 ≤ x ≤ 8 , g 3 ⁡ = 0. … Magnify Figure 23.4.5: σ ⁡ ( x ; g 2 ⁡ , 0 ) for - 5 ≤ x ≤ 5 , g 2 ⁡ = 0. … Magnify
##### 3: 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 $\int{\wp}^{3}\left(z\right)\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-\frac% {3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
##### 4: 23.19 Interrelations
23.19.2 $J\left(\tau\right)=\frac{4}{27}\frac{\left(1-\lambda\left(\tau\right)+{\lambda% }^{2}\left(\tau\right)\right)^{3}}{\left(\lambda\left(\tau\right)\left(1-% \lambda\left(\tau\right)\right)\right)^{2}},$
23.19.3 $J\left(\tau\right)=\frac{{g_{2}}^{3}}{{g_{2}}^{3}-27{g_{3}}^{2}},$
where $g_{2},g_{3}$ are the invariants of the lattice $\mathbb{L}$ with generators $1$ and $\tau$; see §23.3(i). …
##### 5: 23.2 Definitions and Periodic Properties
23.2.4 ${}\wp\left(z\right)=\frac{1}{z^{2}}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right),$
23.2.5 ${}\zeta\left(z\right)=\frac{1}{z}+\sum_{w\in\mathbb{L}\setminus\{0\}}\left(% \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^{2}}\right),$
23.2.6 ${}\sigma\left(z\right)=z\prod_{w\in\mathbb{L}\setminus\{0\}}\left(\left(1-% \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^{2}}{2w^{2}}\right)\right).$
23.2.7 $\wp\left(z\right)=-\zeta'\left(z\right),$
23.2.8 $\zeta\left(z\right)=\ifrac{\sigma'\left(z\right)}{\sigma\left(z\right)}.$
##### 6: 23.10 Addition Theorems and Other Identities
23.10.1 $\wp\left(u+v\right)=\frac{1}{4}\left(\frac{\wp'\left(u\right)-\wp'\left(v% \right)}{\wp\left(u\right)-\wp\left(v\right)}\right)^{2}-\wp\left(u\right)-\wp% \left(v\right),$
23.10.2 $\zeta\left(u+v\right)=\zeta\left(u\right)+\zeta\left(v\right)+\frac{1}{2}\frac% {\zeta''\left(u\right)-\zeta''\left(v\right)}{\zeta'\left(u\right)-\zeta'\left% (v\right)},$
23.10.3 $\frac{\sigma\left(u+v\right)\sigma\left(u-v\right)}{{\sigma}^{2}\left(u\right)% {\sigma}^{2}\left(v\right)}=\wp\left(v\right)-\wp\left(u\right),$
23.10.10 $\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).$
Also, when $\mathbb{L}$ is replaced by $c\mathbb{L}$ the lattice invariants $g_{2}$ and $g_{3}$ are divided by $c^{4}$ and $c^{6}$, respectively. …
##### 7: 23.9 Laurent and Other Power Series
$c_{2}=\frac{1}{20}g_{2},$
$c_{3}=\frac{1}{28}g_{3},$
23.9.7 $\sigma\left(z\right)=\sum_{m,n=0}^{\infty}a_{m,n}(10c_{2})^{m}(56c_{3})^{n}% \frac{z^{4m+6n+1}}{(4m+6n+1)!},$
##### 8: 23.23 Tables
Abramowitz and Stegun (1964) also includes other tables to assist the computation of the Weierstrass functions, for example, the generators as functions of the lattice invariants $g_{2}$ and $g_{3}$. …
##### 9: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … 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 $\eta\left(\tau\right)$. …
##### 10: 23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions $\lambda$, $J$, and $\eta$. … Figure 23.16.1: Modular functions λ ⁡ ( i ⁢ y ) , J ⁡ ( i ⁢ y ) , η ⁡ ( i ⁢ y ) for 0 ≤ y ≤ 3 . … Magnify