elliptic cases

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1: 23.4 Graphics

… ►

§23.4(i) Real Variables

… ►

See accompanying text — Figure 23.4.6: $\sigma\left(x;0,g_{3}\right)$ for $-5\leq x\leq 5$ , $g_{3}$ = 0. … Magnify

►

…

2: 19.16 Definitions

… ►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function … ►

§19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

… ►All other elliptic cases are integrals of the second kind. …

3: 22.21 Tables

… ►Spenceley and Spenceley (1947) tabulates

\operatorname{sn}\left(Kx,k\right)

\operatorname{cn}\left(Kx,k\right)

\operatorname{dn}\left(Kx,k\right)

\operatorname{am}\left(Kx,k\right)

\mathcal{E}\left(Kx,k\right)

for

\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}

and

x=0\left(\tfrac{1}{90}\right)1

to 12D, or 12 decimals of a radian in the case of

\operatorname{am}\left(Kx,k\right)

. …

4: 23.5 Special Lattices

… ►

§23.5(iii) Lemniscatic Lattice

… ►

§23.5(iv) Rhombic Lattice

… ►As a function of

\Im e_{3}

the root

e_{1}

is increasing. For the case

\omega_{3}=e^{\pi i/3}\omega_{1}

see §23.5(v). ►

§23.5(v) Equianharmonic Lattice

…

5: 22.5 Special Values

… ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. … ►For values of

K,K^{\prime}

when

k^{2}=\frac{1}{2}

(lemniscatic case) see §23.5(iii), and for

k^{2}=e^{\mathrm{i}\pi/3}

(equianharmonic case) see §23.5(v).

6: 19.20 Special Cases

§19.20 Special Cases

… ►The general lemniscatic case is … ►where

x, y, z

may be permuted. … ►The general lemniscatic case is … ►

7: 19.21 Connection Formulas

… ►

19.21.15

pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right),

pq=yz

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.21.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

8: 19.7 Connection Formulas

… ►

§19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$

… ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of

\Pi\left(\phi,\alpha^{2},k\right)