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##### 1: 22.6 Elementary Identities
22.6.4 $-k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right)=k^{2}({% \operatorname{cd}}^{2}\left(z,k\right)-1)={k^{\prime}}^{2}(1-{\operatorname{nd% }}^{2}\left(z,k\right)).$
22.6.8 $\operatorname{cd}\left(2z,k\right)=\frac{{\operatorname{cd}}^{2}\left(z,k% \right)-{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right){\operatorname{% nd}}^{2}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(% z,k\right)},$
22.6.9 $\operatorname{sd}\left(2z,k\right)=\frac{2\operatorname{sd}\left(z,k\right)% \operatorname{cd}\left(z,k\right)\operatorname{nd}\left(z,k\right)}{1+k^{2}{k^% {\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k\right)},$
22.6.10 $\operatorname{nd}\left(2z,k\right)=\frac{{\operatorname{nd}}^{2}\left(z,k% \right)+k^{2}{\operatorname{sd}}^{2}\left(z,k\right){\operatorname{cd}}^{2}% \left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k% \right)},$
22.6.21 ${\operatorname{dn}}^{2}\left(\tfrac{1}{2}z,k\right)=\frac{k^{2}\operatorname{% cn}\left(z,k\right)+\operatorname{dn}\left(z,k\right)+{k^{\prime}}^{2}}{1+% \operatorname{dn}\left(z,k\right)}=\frac{{k^{\prime}}^{2}(1-\operatorname{cn}% \left(z,k\right))}{\operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z,% k\right)}=\frac{{k^{\prime}}^{2}(1+\operatorname{dn}\left(z,k\right))}{{k^{% \prime}}^{2}+\operatorname{dn}\left(z,k\right)-k^{2}\operatorname{cn}\left(z,k% \right)}.$
##### 2: 19.4 Derivatives and Differential Equations
$\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}{\mathrm{d}k% }=kK\left(k\right),$
Let $D_{k}=\ifrac{\partial}{\partial k}$. Then …If $\phi=\pi/2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. An analogous differential equation of third order for $\Pi\left(\phi,\alpha^{2},k\right)$ is given in Byrd and Friedman (1971, 118.03).
##### 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 $\operatorname{am}\left(Kx,k\right)$. Curtis (1964b) tabulates $\operatorname{sn}\left(mK/n,k\right)$, $\operatorname{cn}\left(mK/n,k\right)$, $\operatorname{dn}\left(mK/n,k\right)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. Lawden (1989, pp. 280–284 and 293–297) tabulates $\operatorname{sn}\left(x,k\right)$, $\operatorname{cn}\left(x,k\right)$, $\operatorname{dn}\left(x,k\right)$, $\mathcal{E}\left(x,k\right)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. … Zhang and Jin (1996, p. 678) tabulates $\operatorname{sn}\left(Kx,k\right)$, $\operatorname{cn}\left(Kx,k\right)$, $\operatorname{dn}\left(Kx,k\right)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …
##### 4: 22.17 Moduli Outside the Interval [0,1]
$k_{1}k_{1}^{\prime}=\frac{k}{1+k^{2}},$
In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in $k$. … When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^{2}$. …In consequence, the formulas in this chapter remain valid when $k$ is complex. In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of $k$, irrespective of which values of $\sqrt{k}$ and $k^{\prime}=\sqrt{1-k^{2}}$ are chosen—as long as they are used consistently. …
##### 5: 22.7 Landen Transformations
22.7.1 $k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},$
22.7.2 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},$
$k_{2}=\frac{2\sqrt{k}}{1+k},$
$k_{2}^{\prime}=\frac{1-k}{1+k},$
22.7.8 $\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.$
##### 6: 22.13 Derivatives and Differential Equations
(The modulus $k$ is suppressed throughout the table.) …
22.13.2 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cn}\left(z,k\right)\right)^{% 2}={\left(1-{\operatorname{cn}}^{2}\left(z,k\right)\right)}{\left({k^{\prime}}% ^{2}+k^{2}{\operatorname{cn}}^{2}\left(z,k\right)\right)},$
22.13.14 $\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{cn}\left(z,k\right)=-(% {k^{\prime}}^{2}-k^{2})\operatorname{cn}\left(z,k\right)-2k^{2}{\operatorname{% cn}}^{3}\left(z,k\right),$
22.13.17 $\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{sd}\left(z,k\right)=(k% ^{2}-{k^{\prime}}^{2})\operatorname{sd}\left(z,k\right)-2k^{2}{k^{\prime}}^{2}% {\operatorname{sd}}^{3}\left(z,k\right),$
22.13.20 $\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{nc}\left(z,k\right)=(k% ^{2}-{k^{\prime}}^{2})\operatorname{nc}\left(z,k\right)+2{k^{\prime}}^{2}{% \operatorname{nc}}^{3}\left(z,k\right),$
##### 7: 22.5 Special Values
For example, at $z=K+iK^{\prime}$, $\operatorname{sn}\left(z,k\right)=1/k$, $\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0$. (The modulus $k$ is suppressed throughout the table.) … 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 $z$. …
###### §22.5(ii) Limiting Values of $k$
Expansions for $K,K^{\prime}$ as $k\to 0$ or $1$ are given in §§19.5, 19.12. …
##### 8: 22.16 Related Functions
$\operatorname{am}\left(x,k\right)$ is an infinitely differentiable function of $x$. …
###### Approximations for Small $k$, $k^{\prime}$
In Equations (22.16.21)–(22.16.23), $-KIn Equations (22.16.24)–(22.16.26), $-2K. … For $E\left(k\right)$ see §19.2(ii). …
##### 9: 26.1 Special Notation
 $x$ real variable. … $j$ divides $k$. greatest common divisor of positive integers $h$ and $k$.
 $\genfrac{(}{)}{0.0pt}{}{m}{n}$ binomial coefficient. … number of partitions of $n$ into at most $k$ parts. …
Other notations for $s\left(n,k\right)$, the Stirling numbers of the first kind, include $S_{n}^{(k)}$ (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), $S_{n}^{k}$ (Jordan (1939), Moser and Wyman (1958a)), $\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}$ (Milne-Thomson (1933)), $(-1)^{n-k}S_{1}(n-1,n-k)$ (Carlitz (1960), Gould (1960)), $(-1)^{n-k}\left[n\atop k\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). Other notations for $S\left(n,k\right)$, the Stirling numbers of the second kind, include $\mathscr{S}^{(k)}_{n}$ (Fort (1948)), $\mathfrak{S}_{n}^{k}$ (Jordan (1939)), $\sigma_{n}^{k}$ (Moser and Wyman (1958b)), $\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_{2}(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{n\atop k\right\}$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
##### 10: 22.10 Maclaurin Series
22.10.1 $\operatorname{sn}\left(z,k\right)=z-\left(1+k^{2}\right)\frac{z^{3}}{3!}+\left% (1+14k^{2}+k^{4}\right)\frac{z^{5}}{5!}-\left(1+135k^{2}+135k^{4}+k^{6}\right)% \frac{z^{7}}{7!}+O\left(z^{9}\right),$
The full expansions converge when $|z|<\min\left(K\left(k\right),{K^{\prime}}\left(k\right)\right)$.
###### §22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$
The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime}$-plane in the case of (22.10.7)–(22.10.9); see §22.17.