Weierstrass%E2%80%99s

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21: 23.20 Mathematical Applications

§23.20 Mathematical Applications

►

§23.20(i) Conformal Mappings

… ► ►

§23.20(ii) Elliptic Curves

… ►

§23.20(iii) Factorization

…

22: 23.21 Physical Applications

§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. …

23: 19.25 Relations to Other Functions

… ►

§19.25(vi) Weierstrass Elliptic Functions

… ►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)). …

24: 23.23 Tables

§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}

. ►For earlier tables related to Weierstrass functions see Fletcher et al. (1962, pp. 503–505) and Lebedev and Fedorova (1960, pp. 223–226).

25: 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

. …

26: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

23 Weierstrass Elliptic and Modular Functions

…

27: 23.12 Asymptotic Approximations

§23.12 Asymptotic Approximations

… ►

23.12.1

\wp\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(-\frac{1}{3}+{\csc}^{2}% \left(\frac{\pi z}{2\omega_{1}}\right)+8\left(1-\cos\left(\frac{\pi z}{\omega_% {1}}\right)\right)q^{2}+O\left(q^{4}\right)\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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators
Referenced by:: §23.12
Permalink:: http://dlmf.nist.gov/23.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.12 and Ch.23

►

23.12.2

\zeta\left(z\right)=\frac{\pi^{2}}{4\omega_{1}^{2}}\left(\frac{z}{3}+\frac{2% \omega_{1}}{\pi}\cot\left(\frac{\pi z}{2\omega_{1}}\right)-8\left(z-\frac{% \omega_{1}}{\pi}\sin\left(\frac{\pi z}{\omega_{1}}\right)\right)q^{2}+O\left(q% ^{4}\right)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators
Referenced by:: §23.12, Erratum (V1.0.23) for Equation (23.12.2)
Permalink:: http://dlmf.nist.gov/23.12.E2
Encodings:: pMML, png, TeX
Correction (effective with 1.0.23):: The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.
Suggested 2019-05-27 by Blagoje Oblak
See also:: Annotations for §23.12 and Ch.23

►

23.12.3

\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp\left(\frac{\pi^{2}z^{2}}{24% \omega_{1}^{2}}\right)\sin\left(\frac{\pi z}{2\omega_{1}}\right)\*\left(1-% \left(\frac{\pi^{2}z^{2}}{\omega_{1}^{2}}-4{\sin}^{2}\left(\frac{\pi z}{2% \omega_{1}}\right)\right)q^{2}+O\left(q^{4}\right)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $q$ : nome, $z$ : complex and $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators
Permalink:: http://dlmf.nist.gov/23.12.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.12 and Ch.23

…

28: 23.8 Trigonometric Series and Products

… ►

§23.8(i) Fourier Series

… ►

§23.8(ii) Series of Cosecants and Cotangents

… ►

23.8.3

\wp\left(z\right)=-\frac{\eta_{1}}{\omega_{1}}+\frac{\pi^{2}}{4\omega_{1}^{2}}% \sum_{n=-\infty}^{\infty}{\csc}^{2}\left(\frac{\pi(z+2n\omega_{3})}{2\omega_{1% }}\right),

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\mathbb{L}$ : lattice, $n$ : integer, $z$ : complex, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and $\eta_{j}$ : complex number
Permalink:: http://dlmf.nist.gov/23.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.8(ii), §23.8 and Ch.23

… ►where in (23.8.4) the terms in

n

and

-n

are to be bracketed together (the Eisenstein convention or principal value: see Weil (1999, p. 6) or Walker (1996, p. 3)). … ►

§23.8(iii) Infinite Products

…

29: 25.1 Special Notation

… ► ►►►►

$k, m, n$	nonnegative integers.
…
$s=\sigma+it$	complex variable.
…
$\gamma$	Euler’s constant (§5.2(ii)).
…

►The main function treated in this chapter is the Riemann zeta function

\zeta\left(s\right)

. … ►The main related functions are the Hurwitz zeta function

\zeta\left(s,a\right)

, the dilogarithm

\operatorname{Li}_{2}\left(z\right)

, the polylogarithm

\operatorname{Li}_{s}\left(z\right)

(also known as Jonquière’s function

\phi\left(z,s\right)

), Lerch’s transcendent

\Phi\left(z,s,a\right)

, and the Dirichlet

L

-functions

L\left(s,\chi\right)

30: 19.10 Relations to Other Functions

… ►

§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

19.10.2

(\sinh\phi)R_{C}\left(1,{\cosh}^{2}\phi\right)=\operatorname{gd}\left(\phi% \right).

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{gd}\NVar{x}$ : Gudermannian function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $\phi$ : real or complex argument
Referenced by:: §19.10(ii), §19.6(ii)
Permalink:: http://dlmf.nist.gov/19.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.10(ii), §19.10 and Ch.19