Weierstrass P-function

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21: 23 Weierstrass Elliptic and Modular
Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

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22: 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)). …

23: 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). …

24: 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 $d=g_3=4(4-2\mathrm{i})$. …

25: Peter L. Walker

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23 Weierstrass Elliptic and Modular Functions

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26: 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. …

27: 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 $L(s,\chi )$.

28: William P. Reinhardt

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23 Weierstrass Elliptic and Modular Functions

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29: 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. …

30: 14.21 Definitions and Basic Properties

… ►Standard solutions: the associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $P_{\nu }^{-\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{-\nu -1}^{\mu }\left(z\right)$. $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). The principal branches of $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are real when $\nu$, $\mu \in ℝ$ and $z\in (1,\mathrm{\infty })$. … ►The generating function expansions (14.7.19) (with $𝖯$ replaced by $P$) and (14.7.22) apply when ; (14.7.21) (with $𝖯$ replaced by $P$) applies when $|h|>\max \left|z\pm {\left(z^2-1\right)}^{1/2}\right|$.