# lattice

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##### 1: 23.3 Differential Equations
The lattice invariants are defined by … The lattice roots satisfy the cubic equation …and are denoted by $e_{1},e_{2},e_{3}$. … Let ${g_{2}}^{3}\neq 27{g_{3}}^{2}$, or equivalently $\Delta$ be nonzero, or $e_{1},e_{2},e_{3}$ be distinct. … 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. …
##### 2: 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 $\int{\wp}^{3}\left(z\right)\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-\frac% {3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
##### 3: 23.7 Quarter Periods
23.7.1 $\wp\left(\tfrac{1}{2}\omega_{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}% =e_{1}+\omega_{1}^{-2}(K\left(k\right))^{2}k^{\prime},$
23.7.2 $\wp\left(\tfrac{1}{2}\omega_{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})% }=e_{2}-i\omega_{1}^{-2}(K\left(k\right))^{2}kk^{\prime},$
23.7.3 $\wp\left(\tfrac{1}{2}\omega_{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}% =e_{3}-\omega_{1}^{-2}(K\left(k\right))^{2}k,$
where $k,k^{\prime}$ and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.
##### 4: 23.10 Addition Theorems and Other Identities
23.10.4 $\sigma\left(u+v\right)\sigma\left(u-v\right)\sigma\left(x+y\right)\sigma\left(% x-y\right)+\sigma\left(v+x\right)\sigma\left(v-x\right)\sigma\left(u+y\right)% \sigma\left(u-y\right)+{\sigma\left(x+u\right)\sigma\left(x-u\right)\sigma% \left(v+y\right)\sigma\left(v-y\right)=0.}$
For further addition-type identities for the $\sigma$-function see Lawden (1989, §6.4). …
23.10.8 $(\wp\left(2z\right)-e_{1}){\wp'}^{2}(z)=\left((\wp\left(z\right)-e_{1})^{2}-(e% _{1}-e_{2})(e_{1}-e_{3})\right)^{2}.$
(23.10.8) continues to hold when $e_{1}$, $e_{2}$, $e_{3}$ are permuted cyclically. … 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. …
##### 5: 23.2 Definitions and Periodic Properties
###### §23.2(i) Lattices
$\wp\left(z\right)$ and $\zeta\left(z\right)$ are meromorphic functions with poles at the lattice points. $\wp\left(z\right)$ is even and $\zeta\left(z\right)$ is odd. …The function $\sigma\left(z\right)$ is entire and odd, with simple zeros at the lattice points. …
##### 6: 23.4 Graphics
###### §23.4(i) Real Variables
Line graphs of the Weierstrass functions $\wp\left(x\right)$, $\zeta\left(x\right)$, and $\sigma\left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … Figure 23.4.7: ℘ ⁡ ( x ) with ω 1 = K ⁡ ( k ) , ω 3 = i ⁢ K ′ ⁡ ( k ) for 0 ≤ x ≤ 9 , k 2 = 0. … Magnify Surfaces for the Weierstrass functions $\wp\left(z\right)$, $\zeta\left(z\right)$, and $\sigma\left(z\right)$. … Figure 23.4.8: ℘ ⁡ ( x + i ⁢ y ) with ω 1 = K ⁡ ( k ) , ω 3 = i ⁢ K ′ ⁡ ( k ) for - 2 ⁢ K ⁡ ( k ) ≤ x ≤ 2 ⁢ K ⁡ ( k ) , 0 ≤ y ≤ 6 ⁢ K ′ ⁡ ( k ) , k 2 = 0.9 . (The scaling makes the lattice appear to be square.) Magnify 3D Help
##### 7: 23.9 Laurent and Other Power Series
Let $z_{0}(\neq 0)$ be the nearest lattice point to the origin, and define …
$c_{2}=\frac{1}{20}g_{2},$
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 $1/\wp\left(z\right)\to 0$. …
##### 9: 23.23 Tables
2 in Abramowitz and Stegun (1964) gives values of $\wp\left(z\right)$, $\wp'\left(z\right)$, and $\zeta\left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that $\omega_{1}=1$ and $\omega_{3}=ia$ (rectangular case), or $\omega_{1}=1$ and $\omega_{3}=\tfrac{1}{2}+ia$ (rhombic case), for $a$ = 1. …05, and in the case of $\wp\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). 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 $g_{3}$. …
##### 10: 23.6 Relations to Other Functions
In this subsection $2\omega_{1}$, $2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$, and the lattice roots $e_{1}$, $e_{2}$, $e_{3}$ are given by (23.3.9). … For further results for the $\sigma$-function see Lawden (1989, §6.2). … Again, in Equations (23.6.16)–(23.6.26), $2\omega_{1},2\omega_{3}$ are any pair of generators of the lattice $\mathbb{L}$ and $e_{1},e_{2},e_{3}$ are given by (23.3.9). …