Weierstrass M-test

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21: 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)). …

22: 29.2 Differential Equations

… ►we have ►

29.2.9

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\eta}^{2}}+(g-\nu(\nu+1)\wp\left(\eta% \right))w=0,

ⓘ

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\nu$ : real parameter, $g$ and $\eta$
Permalink:: http://dlmf.nist.gov/29.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §29.2(ii), §29.2 and Ch.29

… ►

29.2.11

\zeta=\wp\left(\eta;g_{2},g_{3}\right)=\wp\left(\eta\right),

ⓘ

Defines:: $\zeta$ (locally)
Symbols:: $\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$ : Weierstrass $\wp$ -function, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\eta$ , $g_{2}$ and $g_{3}$
Permalink:: http://dlmf.nist.gov/29.2.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §29.2(ii), §29.2 and Ch.29

… ►For the Weierstrass function

\wp

see §23.2(ii). …

23: 23.22 Methods of Computation

§23.22 Methods of Computation

… ►

§23.22(ii) Lattice Calculations

… ►The corresponding values of

e_{1}

e_{2}

e_{3}

are calculated from (23.6.2)–(23.6.4), then

g_{2}

and

g_{3}

are obtained from (23.3.6) and (23.3.7). … ►Suppose that the invariants

g_{2}=c

g_{3}=d

, are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). … ►Assume

c=g_{2}=-4(3-2i)

and

d=g_{3}=4(4-2i)

. …

24: Peter L. Walker

… ►

23 Weierstrass Elliptic and Modular Functions

…

25: 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. …

26: 25.1 Special Notation

… ►The main related functions are the Hurwitz zeta function

\zeta\left(s,a\right)

, the dilogarithm

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

, the polylogarithm

\mathrm{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)

27: William P. Reinhardt

… ►

23 Weierstrass Elliptic and Modular Functions

…

28: 20.9 Relations to Other Functions

… ►

§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. …

29: Bibliography E

… ►

U. Eckhardt (1980) Algorithm 549: Weierstrass’ elliptic functions. ACM Trans. Math. Software 6 (1), pp. 112–120.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 18D.
External Links:: ISSN 0098-3500, Document, GAMS (module 549; package TOMS)
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. Eichler and D. Zagier (1982) On the zeros of the Weierstrass

\wp

-function. Math. Ann. 258 (4), pp. 399–407.

ⓘ

External Links:: ISSN 0025-5831, Document, MathReview (Toyokazu Hiramatsu), ZentralBlatt
Referenced by:: §23.13.
Encodings:: BibTeX

…

30: 22.1 Special Notation

… ► …