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1: 22.6 Elementary Identities

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22.6.4 $$-k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k)=k^2({\mathrm{cd}}^2(z,k)-1)=k_{}^{\prime }{}_{}{}^2(1-{\mathrm{nd}}^2(z,k)).$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.9.2

Permalink:

http://dlmf.nist.gov/22.6.E4

Encodings:

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See also:

Annotations for §22.6(i), §22.6 and Ch.22

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22.6.8 $$\mathrm{cd}(2z,k)=\frac{{\mathrm{cd}}^2(z,k)-k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k){\mathrm{nd}}^2(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E8

Encodings:

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See also:

Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.9 $$\mathrm{sd}(2z,k)=\frac{2\mathrm{sd}(z,k)\mathrm{cd}(z,k)\mathrm{nd}(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.10 $$\mathrm{nd}(2z,k)=\frac{{\mathrm{nd}}^2(z,k)+k^2{\mathrm{sd}}^2(z,k){\mathrm{cd}}^2(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E10

Encodings:

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See also:

Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.21 $${\mathrm{dn}}^2(\frac{1}{2}z,k)=\frac{k^2\mathrm{cn}(z,k)+\mathrm{dn}(z,k)+k_{}^{\prime }{}_{}{}^2}{1+\mathrm{dn}(z,k)}=\frac{k_{}^{\prime }{}_{}{}^2(1-\mathrm{cn}(z,k))}{\mathrm{dn}(z,k)-\mathrm{cn}(z,k)}=\frac{k_{}^{\prime }{}_{}{}^2(1+\mathrm{dn}(z,k))}{k_{}^{\prime }{}_{}{}^2+\mathrm{dn}(z,k)-k^2\mathrm{cn}(z,k)}.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.19.3

Permalink:

http://dlmf.nist.gov/22.6.E21

Encodings:

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See also:

Annotations for §22.6(iii), §22.6 and Ch.22

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2: 19.4 Derivatives and Differential Equations

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${\displaystyle \frac{d(E\left(k\right)-k_{}^{\prime }{}_{}{}^{2}K\left(k\right))}{dk}}=kK\left(k\right),$

… ►Let $D_k=\partial /\partial k$. Then …If $\varphi =\pi /2$, 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 $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is given in Byrd and Friedman (1971, 118.03).

3: 22.21 Tables

… ►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. ►Curtis (1964b) tabulates $\mathrm{sn}(mK/n,k)$, $\mathrm{cn}(mK/n,k)$, $\mathrm{dn}(mK/n,k)$ 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 $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ 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 $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …

4: 22.17 Moduli Outside the Interval [0,1]

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$k_1{k}_1^{\prime }={\displaystyle \frac{k}{1+k^2}},$

… ►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

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22.7.1 $$k_1=\frac{1-k^{\prime }}{1+k^{\prime }},$$

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Defines:

$k_1$: change of variable (locally)

Symbols:

$k^{\prime }$: complementary modulus

A&S Ref:

16.12.1

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.7.E1

Encodings:

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See also:

Annotations for §22.7(i), §22.7 and Ch.22

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22.7.2 $$\mathrm{sn}(z,k)=\frac{(1+k_1)\mathrm{sn}(z/(1+k_1),k_1)}{1+k_1{\mathrm{sn}}^2(z/(1+k_1),k_1)},$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k_1$: change of variable

A&S Ref:

16.12.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.7.E2

Encodings:

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See also:

Annotations for §22.7(i), §22.7 and Ch.22

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$k_2={\displaystyle \frac{2\sqrt{k}}{1+k}},$

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$k_2^{\prime }={\displaystyle \frac{1-k}{1+k}},$

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22.7.8 $$\mathrm{dn}(z,k)=\frac{(1-k_2^{\prime })({\mathrm{dn}}^2(z/(1+k_2^{\prime }),k_2)+k_2^{\prime })}{k_2^2\mathrm{dn}(z/(1+k_2^{\prime }),k_2)}.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.4

Permalink:

http://dlmf.nist.gov/22.7.E8

Encodings:

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See also:

Annotations for §22.7(ii), §22.7 and Ch.22

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6: 22.13 Derivatives and Differential Equations

… ►(The modulus $k$ is suppressed throughout the table.) … ►

22.13.2 $${\left(\frac{d}{dz}\mathrm{cn}(z,k)\right)}^2=\left(1-{\mathrm{cn}}^2(z,k)\right)\left(k_{}^{\prime }{}_{}{}^2+k^2{\mathrm{cn}}^2(z,k)\right),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E2

Encodings:

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See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.14 $$\frac{d^2}{{dz}^2}\mathrm{cn}(z,k)=-(k_{}^{\prime }{}_{}{}^2-k^2)\mathrm{cn}(z,k)-2k^2{\mathrm{cn}}^3(z,k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Referenced by:

§22.10(i), §22.10(ii)

Permalink:

http://dlmf.nist.gov/22.13.E14

Encodings:

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See also:

Annotations for §22.13(iii), §22.13 and Ch.22

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22.13.17 $$\frac{d^2}{{dz}^2}\mathrm{sd}(z,k)=(k^2-k_{}^{\prime }{}_{}{}^2)\mathrm{sd}(z,k)-2k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^3(z,k),$$

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Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E17

Encodings:

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See also:

Annotations for §22.13(iii), §22.13 and Ch.22

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22.13.20 $$\frac{d^2}{{dz}^2}\mathrm{nc}(z,k)=(k^2-k_{}^{\prime }{}_{}{}^2)\mathrm{nc}(z,k)+2k_{}^{\prime }{}_{}{}^2{\mathrm{nc}}^3(z,k),$$

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Symbols:

$\mathrm{nc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E20

Encodings:

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See also:

Annotations for §22.13(iii), §22.13 and Ch.22

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7: 22.5 Special Values

… ►For example, at $z=K+\mathrm{i}{K}^{\prime }$, $\mathrm{sn}(z,k)=1/k$, $d\mathrm{sn}(z,k)/dz=0$. (The modulus $k$ is suppressed throughout the table.) … ►Table 22.5.2 gives $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ 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

… ► $\mathrm{am}(x,k)$ is an infinitely differentiable function of $x$. … ►

Approximations for Small $k$, $k^{\prime }$

… ►In Equations (22.16.21)–(22.16.23), … ►In Equations (22.16.24)–(22.16.26), . … ►For $E\left(k\right)$ see §19.2(ii). …

9: 26.1 Special Notation

… ► ►►►►

$x$	real variable.
…
$j|k$	$j$ divides $k$.
$\left(h,k\right)$	greatest common divisor of positive integers $h$ and $k$.

… ► ►►►

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
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$p_k\left(n\right)$	number of partitions of $n$ into at most $k$ parts.
…

… ►Other notations for $s(n,k)$, 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)), $\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)B_{n-k}^{(n)}$ (Milne-Thomson (1933)), ${(-1)}^{n-k}{S}_1(n-1,n-k)$ (Carlitz (1960), Gould (1960)), ${(-1)}^{n-k}\left[\genfrac{}{}{0pt}{}{n}{k}\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for $S(n,k)$, the Stirling numbers of the second kind, include ${𝒮}_n^{(k)}$ (Fort (1948)), ${𝔖}_n^k$ (Jordan (1939)), ${\sigma }_n^k$ (Moser and Wyman (1958b)), $\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_2(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{\genfrac{}{}{0pt}{}{n}{k}\right\}$ (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

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22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

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See also:

Annotations for §22.10(i), §22.10 and Ch.22

… ►The full expansions converge when . ►

§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.