Jacobi elliptic form

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1: 31.2 Differential Equations

… ►

Jacobi’s Elliptic Form

…

2: 22.5 Special Values

… ►For example, at

z=K+iK^{\prime}

\operatorname{sn}\left(z,k\right)=1/k

\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0

. … ►Table 22.5.2 gives

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

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

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

for other special values of

z

. For example,

\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}

. … ►

§22.5(ii) Limiting Values of $k$

… ►

3: 22.18 Mathematical Applications

… ►For any two points

(x_{1},y_{1})

and

(x_{2},y_{2})

on this curve, their sum

(x_{3},y_{3})

, always a third point on the curve, is defined by the Jacobi–Abel addition law …

4: Errata

… ►

Table 22.5.4

Originally the limiting form for $\operatorname{sc}\left(z,k\right)$ in the last line of this table was incorrect ( $\cosh z$ , instead of $\sinh z$ ).

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\tanh z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\coth z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$
$\operatorname{dn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{nd}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{sc}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{cs}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$

Reported 2010-11-23.

…

5: 29.12 Definitions

… ►With the substitution

\xi={\operatorname{sn}}^{2}\left(z,k\right)

every Lamé polynomial in Table 29.12.1 can be written in the form …

6: 29.15 Fourier Series and Chebyshev Series

… ►Since (29.2.5) implies that

\cos\phi=\operatorname{sn}\left(z,k\right)

, (29.15.1) can be rewritten in the form …

7: 22.8 Addition Theorems

§22.8 Addition Theorems

… ►

§22.8(ii) Alternative Forms for Sum of Two Arguments

… ►

§22.8(iii) Special Relations Between Arguments

… ►If sums/differences of the

z_{j}

’s are rational multiples of

K\left(k\right)

, then further relations follow. …

8: 29.2 Differential Equations

… ►For

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

see §22.2. … ►

§29.2(ii) Other Forms

… ►For

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

see §22.16(i). … ►we have … ►

9: 22.15 Inverse Functions

§22.15 Inverse Functions

… ►are denoted respectively by … ►

§22.15(ii) Representations as Elliptic Integrals

… ►The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. … ►

10: 22.4 Periods, Poles, and Zeros

… ►For example, the poles of

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

, abbreviated as

\operatorname{sn}

in the following tables, are at

z=2mK+(2n+1)iK^{\prime}

. … ►Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form

2mK+2niK^{\prime}

, where

m,n\in\mathbb{Z}

. … ►Then: (a) In any lattice unit cell

\operatorname{pq}\left(z,k\right)

has a simple zero at

z=\mbox{p}

and a simple pole at

z=\mbox{q}

. (b) The difference between p and the nearest q is a half-period of

\operatorname{pq}\left(z,k\right)

. … ►For example,

\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)

. …