DLMF: Untitled Document

… ►

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},

ⓘ

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, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

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},

ⓘ

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, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►

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,

ⓘ

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, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.7 and Ch.23

►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.

… ►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. …

… ►

23.14.1

\int\wp\left(z\right)\,\mathrm{d}z=-\zeta\left(z\right),

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.6.1
Permalink:: http://dlmf.nist.gov/23.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

►

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,

ⓘ

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 $g_{2}$ , $g_{3}$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.1
Permalink:: http://dlmf.nist.gov/23.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

►

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.

ⓘ

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 $g_{2}$ , $g_{3}$ , $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbb{L}$ : lattice and $z$ : complex
A&S Ref:: 18.7.2
Permalink:: http://dlmf.nist.gov/23.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.14 and Ch.23

…

… ►

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.}

ⓘ

Symbols:: $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\mathbb{L}$ : lattice, $x$ : real part of $z$ and $y$ : imaginary part of $z$
Permalink:: http://dlmf.nist.gov/23.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(i), §23.10 and Ch.23

►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}.

ⓘ

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 $z$ : complex
Referenced by:: §23.10(ii), §23.10(ii)
Permalink:: http://dlmf.nist.gov/23.10.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §23.10(ii), §23.10 and Ch.23

►(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. …

… ►

§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. …

… ►The Weierstrass function

\wp

plays a similar role for cubic potentials in canonical form

g_{3}+g_{2}x-4x^{3}

. … ►Airault et al. (1977) applies the function

\wp

to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … ►where

x, y, z

are the corresponding Cartesian coordinates and

e_{1}

,

e_{2}

,

e_{3}

are constants. … ►

23.21.3

f(\rho)=2\left((\rho-e_{1})(\rho-e_{2})(\rho-e_{3})\right)^{1/2}.

ⓘ

Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\rho$ : roots and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

►Another form is obtained by identifying

e_{1}

,

e_{2}

,

e_{3}

as lattice roots (§23.3(i)), and setting …

… ►

§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. … ►

See accompanying text — Figure 23.4.7: $\wp\left(x\right)$ with $\omega_{1}=K\left(k\right)$ , $\omega_{3}=i{K^{\prime}}\left(k\right)$ for $0\leq x\leq 9$ , $k^{2}$ = 0. … Magnify

… ►Surfaces for the Weierstrass functions

\wp\left(z\right)

,

\zeta\left(z\right)

, and

\sigma\left(z\right)

. … ►

…

… ►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),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $c_{n}$ and $t$ : variable
Referenced by:: §23.9
Permalink:: http://dlmf.nist.gov/23.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §23.9 and Ch.23

… ►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

. …

§23.5 Special Lattices

… ►

§23.5(ii) Rectangular Lattice

… ►

§23.5(iii) Lemniscatic Lattice

… ►

§23.5(iv) Rhombic Lattice

… ►

§23.5(v) Equianharmonic Lattice

…

… ►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}

. …

lattice

1—10 of 58 matching pages

1: 23.7 Quarter Periods

2: 23.3 Differential Equations

3: 23.14 Integrals

4: 23.10 Addition Theorems and Other Identities

5: 23.2 Definitions and Periodic Properties

§23.2(i) Lattices

6: 23.21 Physical Applications

7: 23.4 Graphics

§23.4(i) Real Variables

8: 23.9 Laurent and Other Power Series

9: 23.5 Special Lattices

§23.5 Special Lattices

§23.5(ii) Rectangular Lattice

§23.5(iii) Lemniscatic Lattice

§23.5(iv) Rhombic Lattice

§23.5(v) Equianharmonic Lattice

10: 23.23 Tables