Weierstrass elliptic-function form

SearchAdvancedHelp

(0.005 seconds)

11—20 of 300 matching pages

11: 23.10 Addition Theorems and Other Identities

… ►

§23.10(i) Addition Theorems

… ►

§23.10(ii) Duplication Formulas

… ►(23.10.8) continues to hold when $e_1$, $e_2$, $e_3$ are permuted cyclically. … ►

§23.10(iii) $n$-Tuple Formulas

… ►

§23.10(iv) Homogeneity

…

12: 29.2 Differential Equations

… ►

§29.2(i) Lamé’s Equation

… ►

§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). … ►

13: 23.4 Graphics

… ►

§23.4(i) Real Variables

►Line graphs of the Weierstrass functions $\mathrm{\wp }\left(x\right)$, $\zeta \left(x\right)$, and $\sigma \left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … ►

►►

Figure 23.4.6: $\sigma (x;0,g_3)$ for $-5\le x\le 5$, $g_3$ = 0. … Magnify

… ►

§23.4(ii) Complex Variables

►Surfaces for the Weierstrass functions $\mathrm{\wp }\left(z\right)$, $\zeta \left(z\right)$, and $\sigma \left(z\right)$. …

14: 19.25 Relations to Other Functions

… ►

§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)). …

15: 23.23 Tables

§23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp }\left(z\right)$, ${\mathrm{\wp }}^{\prime }\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega }_1=1$ and ${\omega }_3=\mathrm{i}a$ (rectangular case), or ${\omega }_1=1$ and ${\omega }_3=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1. …05, and in the case of $\mathrm{\wp }\left(z\right)$ 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).

16: 23.1 Special Notation

… ► ►►►►

$𝕃$	lattice in $ℂ$.
…
$={\mathrm{e}}^{\mathrm{i}\pi \tau }$	nome.
$\mathrm{\Delta }$	discriminant $g_2^3-27g_3^2$.
…

►The main functions treated in this chapter are the Weierstrass $\mathrm{\wp }$-function $\mathrm{\wp }\left(z\right)=\mathrm{\wp }\left(z|𝕃\right)=\mathrm{\wp }(z;g_2,g_3)$; the Weierstrass zeta function $\zeta \left(z\right)=\zeta \left(z|𝕃\right)=\zeta (z;g_2,g_3)$; the Weierstrass sigma function $\sigma \left(z\right)=\sigma \left(z|𝕃\right)=\sigma (z;g_2,g_3)$; the elliptic modular function $\lambda \left(\tau \right)$; Klein’s complete invariant $J\left(\tau \right)$; Dedekind’s eta function $\eta \left(\tau \right)$. …

17: 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

… ►

18: 23.5 Special Lattices

… ►

§23.5(ii) Rectangular Lattice

… ►In this case the lattice roots $e_1$, $e_2$, and $e_3$ are real and distinct. … ►

§23.5(iii) Lemniscatic Lattice

… ►

§23.5(iv) Rhombic Lattice

… ►

§23.5(v) Equianharmonic Lattice

…

19: 23.11 Integral Representations

§23.11 Integral Representations

… ►

23.11.2 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+8{\int }_0^{\mathrm{\infty }}s\left({\mathrm{e}}^{-s}{\sinh }^2\left(\frac{1}{2}zs\right)f_1(s,\tau )+{\mathrm{e}}^{\mathrm{i}\tau s}{\sin }^2\left(\frac{1}{2}zs\right)f_2(s,\tau )\right)ds,$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: 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

… ►

23.11.3 $$\zeta \left(z\right)=\frac{1}{z}+{\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{-s}\left(zs-\sinh \left(zs\right)\right)f_1(s,\tau )-{\mathrm{e}}^{\mathrm{i}\tau s}\left(zs-\sin \left(zs\right)\right)f_2(s,\tau )\right)ds,$$

ⓘ

Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $𝕃$: 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

…

20: 23.19 Interrelations

… ►

23.19.3 $$J\left(\tau \right)=\frac{g_2^3}{g_2^3-27g_3^2},$$

ⓘ

Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $𝕃$: lattice and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.19 and Ch.23

►where $g_2,g_3$ are the invariants of the lattice $𝕃$ with generators $1$ and $\tau$; see §23.3(i). …