Weierstrass%20product

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11: 23.20 Mathematical Applications

§23.20 Mathematical Applications

в–є

§23.20(i) Conformal Mappings

… в–є … в–є

§23.20(iii) Factorization

… в–є

§23.20(v) Modular Functions and Number Theory

…

13: 23.21 Physical Applications

§23.21 Physical Applications

… в–єThe Weierstrass function

\wp

plays a similar role for cubic potentials in canonical form

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

. … в–є

§23.21(ii) Nonlinear Evolution Equations

… в–є

§23.21(iii) Ellipsoidal Coordinates

… в–єwhere

x, y, z

are the corresponding Cartesian coordinates and

e_{1}

e_{2}

e_{3}

are constants. …

14: Peter L. Walker

… в–є

20 Theta Functions

… в–є

23 Weierstrass Elliptic and Modular Functions

…

15: 19.25 Relations to Other Functions

… в–є

§19.25(vi) Weierstrass Elliptic Functions

… в–єLet

\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

. … в–єfor some

2\omega_{j}\in\mathbb{L}

and

\wp\left(\omega_{j}\right)=e_{j}

. … в–єin 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)). …

16: 23.23 Tables

§23.23 Tables

… в–є2 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). в–єAbramowitz 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}

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

17: William P. Reinhardt

… в–є

20 Theta Functions

… в–є

23 Weierstrass Elliptic and Modular Functions

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

18: 23.6 Relations to Other Functions

… в–є

§23.6(i) Theta Functions

… в–є

§23.6(ii) Jacobian Elliptic Functions

… в–є

§23.6(iii) General Elliptic Functions

… в–є

§23.6(iv) Elliptic Integrals

… в–є

19: 23.5 Special Lattices

… в–є

§23.5(ii) Rectangular Lattice

… в–єIn this case the lattice roots

e_{1}

e_{2}

, and

e_{3}

are real and distinct. … в–є

§23.5(iii) Lemniscatic Lattice

… в–є

§23.5(iv) Rhombic Lattice

… в–є

§23.5(v) Equianharmonic Lattice

…

20: 23.11 Integral Representations

§23.11 Integral Representations

… в–є

23.11.2

\wp\left(z\right)=\frac{1}{z^{2}}+8\int_{0}^{\infty}s\left(e^{-s}{\sinh}^{2}% \left(\tfrac{1}{2}zs\right)f_{1}(s,\tau)+e^{i\tau s}{\sin}^{2}\left(\tfrac{1}{% 2}zs\right)f_{2}(s,\tau)\right)\,\mathrm{d}s,

в“

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $z$ : complex, $\tau$ : complex variable and $f_{j}(s,\tau)$ : functions
Permalink:: http://dlmf.nist.gov/23.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.11 and Ch.23

… в–є

23.11.3

\zeta\left(z\right)=\frac{1}{z}+\int_{0}^{\infty}\left(e^{-s}\left(zs-\sinh% \left(zs\right)\right)f_{1}(s,\tau)-e^{i\tau s}\left(zs-\sin\left(zs\right)% \right)f_{2}(s,\tau)\right)\,\mathrm{d}s,

в“

Symbols:: $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): Weierstrass zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathbb{L}$ : lattice, $z$ : complex, $\tau$ : complex variable and $f_{j}(s,\tau)$ : functions
Permalink:: http://dlmf.nist.gov/23.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §23.11 and Ch.23

…

Weierstrass%20product

11—20 of 271 matching pages

11: 23.20 Mathematical Applications

§23.20 Mathematical Applications

§23.20(i) Conformal Mappings

§23.20(iii) Factorization

§23.20(v) Modular Functions and Number Theory

12: 23.4 Graphics

§23.4(i) Real Variables

§23.4(ii) Complex Variables

13: 23.21 Physical Applications

§23.21 Physical Applications

§23.21(ii) Nonlinear Evolution Equations

§23.21(iii) Ellipsoidal Coordinates

14: Peter L. Walker

15: 19.25 Relations to Other Functions

§19.25(vi) Weierstrass Elliptic Functions

16: 23.23 Tables

§23.23 Tables

17: William P. Reinhardt

18: 23.6 Relations to Other Functions

§23.6(i) Theta Functions

§23.6(ii) Jacobian Elliptic Functions

§23.6(iii) General Elliptic Functions

§23.6(iv) Elliptic Integrals

19: 23.5 Special Lattices

§23.5(ii) Rectangular Lattice

§23.5(iii) Lemniscatic Lattice

§23.5(iv) Rhombic Lattice

§23.5(v) Equianharmonic Lattice

20: 23.11 Integral Representations

§23.11 Integral Representations