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1: 23.2 Definitions and Periodic Properties

… ►

§23.2(i) Lattices

… ► … ►

§23.2(ii) Weierstrass Elliptic Functions

… ► ►

§23.2(iii) Periodicity

…

2: 1.13 Differential Equations

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§1.13(vii) Closed-Form Solutions

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes

\frac{\mathrm{d}}{\mathrm{d}x}

. Assuming that

u(x)

satisfies un-mixed boundary conditions of the form … ►

Transformation to Liouville normal Form

…

3: 23.14 Integrals

§23.14 Integrals

►

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

…

4: 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),

ⓘ

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

. …

5: 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(ii) Nonlinear Evolution Equations

… ►

§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 Weierstrass Elliptic and Modular
Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

…

7: Peter L. Walker

… ►

20 Theta Functions

… ►

23 Weierstrass Elliptic and Modular Functions

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8: William P. Reinhardt

… ►

20 Theta Functions

… ►

23 Weierstrass Elliptic and Modular Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

9: 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. … ►

§23.3(ii) Differential Equations and Derivatives

…

10: 31.2 Differential Equations

… ►

§31.2(iii) Trigonometric Form

… ►

§31.2(iv) Doubly-Periodic Forms

►

Jacobi’s Elliptic Form

… ►

Weierstrass’s Form