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11: 22.5 Special Values

… ►

Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.

► ►►►

	$z$
…
$\operatorname{cn}z$	$1,0$	$0,-k^{\prime}$	$-ik^{\prime}/k,0$	$\infty$ , $-i/k$	$-1,0$	$1,0$	$-1,0$
…

► ►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

. … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as

k\to 0

► ►►►

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\sin z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$\cos z$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$\sec z$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\csc z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\cos z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sin z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\sec z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\csc z$
…

► ►

Table 22.5.4: Limiting forms of Jacobian elliptic functions as

k\to 1

► ►►►

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

► … ►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). …

12: 31.2 Differential Equations

… ►where

2\omega_{1}

and

2\omega_{3}

with

\Im\left(\omega_{3}/\omega_{1}\right)>0

are generators of the lattice

\mathbb{L}

for

\wp\left(z|\mathbb{L}\right)

. … ►Lastly,

w(z)=(z-a)^{1-\epsilon}w_{3}(z)

satisfies (31.2.1) if

w_{3}

is a solution of (31.2.1) with transformed parameters

q_{3}=q+\gamma(1-\epsilon)

;

\alpha_{3}=\alpha+1-\epsilon

\beta_{3}=\beta+1-\epsilon

\epsilon_{3}=2-\epsilon

. By composing these three steps, there result

2^{3}=8

possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). … ►If

\tilde{z}=\tilde{z}(z)

is one of the

3!=6

homographies that map

\infty

\infty

, then

w(z)=\tilde{w}(\tilde{z})

satisfies (31.2.1) if

\tilde{w}(\tilde{z})

is a solution of (31.2.1) with

z

replaced by

\tilde{z}

and appropriately transformed parameters. …If

\tilde{z}=\tilde{z}(z)

is one of the

4!-3!=18

homographies that do not map

\infty

\infty

, then an appropriate prefactor must be included on the right-hand side. …

13: 22.6 Elementary Identities

… ►

22.6.6

\operatorname{cn}\left(2z,k\right)=\frac{{\operatorname{cn}}^{2}\left(z,k% \right)-{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,% k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{\operatorname% {cn}}^{4}\left(z,k\right)-{k^{\prime}}^{2}{\operatorname{sn}}^{4}\left(z,k% \right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.18.2
Referenced by:: §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

… ►

22.6.17

\frac{1-\operatorname{cn}\left(2z,k\right)}{1+\operatorname{cn}\left(2z,k% \right)}=\frac{{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}% \left(z,k\right)}{{\operatorname{cn}}^{2}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.18.4
Permalink:: http://dlmf.nist.gov/22.6.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

… ►

22.6.19

{\operatorname{sn}}^{2}\left(\tfrac{1}{2}z,k\right)=\frac{1-\operatorname{cn}% \left(z,k\right)}{1+\operatorname{dn}\left(z,k\right)}=\frac{1-\operatorname{% dn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}\left(z,k\right))}=\frac{% \operatorname{dn}\left(z,k\right)-k^{2}\operatorname{cn}\left(z,k\right)-{k^{% \prime}}^{2}}{k^{2}(\operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z% ,k\right))},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.19.1
Permalink:: http://dlmf.nist.gov/22.6.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(iii), §22.6 and Ch.22

►

22.6.20

{\operatorname{cn}}^{2}\left(\tfrac{1}{2}z,k\right)=\frac{-{k^{\prime}}^{2}+% \operatorname{dn}\left(z,k\right)+k^{2}\operatorname{cn}\left(z,k\right)}{k^{2% }(1+\operatorname{cn}\left(z,k\right))}=\frac{{k^{\prime}}^{2}(1-\operatorname% {dn}\left(z,k\right))}{k^{2}(\operatorname{dn}\left(z,k\right)-\operatorname{% cn}\left(z,k\right))}=\frac{{k^{\prime}}^{2}(1+\operatorname{cn}\left(z,k% \right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,k\right)-k^{2}% \operatorname{cn}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.19.2
Permalink:: http://dlmf.nist.gov/22.6.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(iii), §22.6 and Ch.22

►

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)}.

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : 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:: pMML, png, TeX
See also:: Annotations for §22.6(iii), §22.6 and Ch.22

…

14: 22.16 Related Functions

… ►

22.16.13

\operatorname{cn}\left(x,k\right)=\cos\phi=\cos\left(\operatorname{am}\left(x,% k\right)\right).

ⓘ

Symbols:: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cos\NVar{z}$ : cosine function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

… ►

22.16.16

\mathcal{E}\left(x,k\right)=k^{2}\int_{0}^{x}{\operatorname{cn}}^{2}\left(t,k% \right)\,\mathrm{d}t+{k^{\prime}}^{2}x,

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.26.2
Permalink:: http://dlmf.nist.gov/22.16.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

… ►

22.16.24

\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({\operatorname{ns}}^{2}\left(t,% k\right)-t^{-2}\right)\,\mathrm{d}t+x^{-1}+x-\operatorname{cn}\left(x,k\right)% \operatorname{ds}\left(x,k\right),

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus
A&S Ref:: 16.26.10
Referenced by:: §22.16(ii)
Permalink:: http://dlmf.nist.gov/22.16.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

►

22.16.25

\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({\operatorname{ds}}^{2}\left(t,% k\right)-t^{-2}\right)\,\mathrm{d}t+x^{-1}+{k^{\prime}}^{2}x-\operatorname{cn}% \left(x,k\right)\operatorname{ds}\left(x,k\right),

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.26.11
Permalink:: http://dlmf.nist.gov/22.16.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

… ►where

\xi=x/{\theta_{3}}^{2}\left(0,q\right)

. …

15: 29.15 Fourier Series and Chebyshev Series

… ►

Polynomial $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$

►When

\nu=2n+3

m=0,1,\dots,n

, the Fourier series (29.6.53) terminates: … ►

29.15.45

\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \sum_{p=0}^{n}B_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)\right),

ⓘ

Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E45
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

… ►

29.15.47

\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \sum_{p=0}^{n}B_{2p+2}U_{2p+1}\left(\operatorname{sn}\left(z,k\right)\right),

ⓘ

Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E47
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

… ►

29.15.50

\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(% \operatorname{sn}\left(z,k\right)\right).

ⓘ

Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E50
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

…

16: 22.20 Methods of Computation

… ►

\operatorname{cn}\left(x,k\right)=\cos\phi_{0},

… ►To compute

\operatorname{sn}

\operatorname{cn}

\operatorname{dn}

to 10D when

x=0.8

k=0.65

. … ►Then from (22.20.5),

\operatorname{sn}\left(0.8,0.65\right)=0.69506\;42165

\operatorname{cn}\left(0.8,0.65\right)=0.71894\;76580

\operatorname{dn}\left(0.8,0.65\right)=0.89212\;34349

. … ►If needed, the corresponding values of

\operatorname{sn}

and

\operatorname{cn}

can be found subsequently by applying (22.10.4) and (22.7.2), followed by (22.10.5) and (22.7.3). … ►If either

\tau

q=e^{i\pi\tau}

is given, then we use

k={\theta_{2}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)

k^{\prime}={\theta_{4}}^{2}\left(0,q\right)/{\theta_{3}}^{2}\left(0,q\right)

K=\frac{1}{2}\pi{\theta_{3}}^{2}\left(0,q\right)

, and

K^{\prime}=-i\tau K

, obtaining the values of the theta functions as in §20.14. …

17: 29.12 Definitions

… ►

29.12.8

\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\mathit{Es}^{2m+2}_{2n+3}\left(z,% k^{2}\right),

ⓘ

Defines:: $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial
Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $m$ : nonnegative integer, $n$ : nonnegative integer, $z$ : complex variable and $k$ : real parameter
Referenced by:: §29.12(i)
Permalink:: http://dlmf.nist.gov/29.12.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §29.12(i), §29.12 and Ch.29

… ►These functions are polynomials in

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

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

, and

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

. … ►

Table 29.12.1: Lamé polynomials.

► ►►►►►

\nu

eigenvalue

h

eigenfunction

w(z)

polynomial

form

real

period

imag.

period

parity of

w(z)

parity of

w(z-K)

parity of

w(z-K-\mathrm{i}{K^{\prime}})

…

2n+1\;

b^{2m+1}_{\nu}\left(k^{2}\right)

\mathit{cE}^{m}_{\nu}\left(z,k^{2}\right)

\operatorname{cn}P({\operatorname{sn}}^{2})

4K

4\mathrm{i}{K^{\prime}}

even

odd

even

…

2n+2\;

b^{2m+2}_{\nu}\left(k^{2}\right)

\mathit{scE}^{m}_{\nu}\left(z,k^{2}\right)

\operatorname{sn}\operatorname{cn}P({\operatorname{sn}}^{2})

2K

4\mathrm{i}{K^{\prime}}

odd

even

…

2n+3\;

b^{2m+2}_{\nu}\left(k^{2}\right)

\mathit{scdE}^{m}_{\nu}\left(z,k^{2}\right)

\operatorname{sn}\operatorname{cn}\operatorname{dn}P({\operatorname{sn}}^{2})

2K

2\mathrm{i}{K^{\prime}}

odd

► …

18: 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. …

19: 22.2 Definitions

… ►

k=\frac{{\theta_{2}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q\right)},

►

k^{\prime}=\frac{{\theta_{4}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)},

►

K\left(k\right)=\frac{\pi}{2}{\theta_{3}}^{2}\left(0,q\right),

… ►

22.2.5

\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nc}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►and on the left-hand side of (22.2.11)

\mathrm{p}

\mathrm{q}

are any pair of the letters

\mathrm{s}

\mathrm{c}

\mathrm{d}

\mathrm{n}

, and on the right-hand side they correspond to the integers

1,2,3,4

20: 22.7 Landen Transformations

… ►

22.7.3

\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.3
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

… ►

22.7.6

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.2
Permalink:: http://dlmf.nist.gov/22.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

►

22.7.7

\operatorname{cn}\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)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.3
Permalink:: http://dlmf.nist.gov/22.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

…

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11—20 of 579 matching pages

11: 22.5 Special Values

12: 31.2 Differential Equations

13: 22.6 Elementary Identities

14: 22.16 Related Functions

15: 29.15 Fourier Series and Chebyshev Series

Polynomial 𝑠𝑐𝑑𝐸 2 ⁢ n + 3 m ⁡ ( z , k 2 )

16: 22.20 Methods of Computation

17: 29.12 Definitions

18: 22.21 Tables

19: 22.2 Definitions

20: 22.7 Landen Transformations

Polynomial $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$