Weierstrass M-test

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21: 19.25 Relations to Other Functions

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§19.25(vi) Weierstrass Elliptic Functions

… ►Let $𝕃$ be a lattice for the Weierstrass elliptic function $\mathrm{\wp }\left(z\right)$. …The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which , for some $j$. … ►for some $2{\omega }_j\in 𝕃$ and $\mathrm{\wp }\left({\omega }_j\right)=e_j$. … ►in which $2{\omega }_1$ and $2{\omega }_3$ are generators for the lattice $𝕃$, ${\omega }_2=-{\omega }_1-{\omega }_3$, and ${\eta }_j=\zeta \left({\omega }_j\right)$ (see (23.2.12)). …

22: 29.2 Differential Equations

… ►we have ►

29.2.9 $$\frac{d^{2}w}{{d\eta }^2}+(g-\nu (\nu +1)\mathrm{\wp }\left(\eta \right))w=0,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: 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

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29.2.11 $$\zeta =\mathrm{\wp }(\eta ;g_2,g_3)=\mathrm{\wp }\left(\eta \right),$$

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Defines:

$\zeta$ (locally)

Symbols:

$\mathrm{\wp }(z;g_2,g_3)$: Weierstrass $\mathrm{\wp }$-function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-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 $\mathrm{\wp }$ see §23.2(ii). …

23: 23.22 Methods of Computation

§23.22 Methods of Computation

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§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 $c=g_2=-4(3-2\mathrm{i})$ and $d=g_3=4(4-2\mathrm{i})$. …

24: Peter L. Walker

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23 Weierstrass Elliptic and Modular Functions

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25: 19.10 Relations to Other Functions

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§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 $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

27: William P. Reinhardt

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23 Weierstrass Elliptic and Modular Functions

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28: 20.9 Relations to Other Functions

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§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

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

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Notes:

Single-precision Fortran, maximum accuracy 18D.

External Links:

ISSN 0098-3500, Document, GAMS (module 549; package TOMS)

Referenced by:

1st item.

Encodings:

BibTeX

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On the zeros of the Weierstrass $\mathrm{\wp }$-function. Math. Ann. 258 (4), pp. 399–407.

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External Links:

ISSN 0025-5831, Document, MathReview (Toyokazu Hiramatsu), ZentralBlatt

Referenced by:

§23.13.

Encodings:

BibTeX

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30: 22.1 Special Notation

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