$\operatorname{sn}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sc}\left(z,k^{\prime}\right)$	$\operatorname{dc}\left(iz,k\right)$ $\;=\;$	$\operatorname{dn}\left(z,k^{\prime}\right)$
$\operatorname{cn}\left(iz,k\right)$ $\;=\;$	$\operatorname{nc}\left(z,k^{\prime}\right)$	$\operatorname{nc}\left(iz,k\right)$ $\;=\;$	$\operatorname{cn}\left(z,k^{\prime}\right)$
$\operatorname{dn}\left(iz,k\right)$ $\;=\;$	$\operatorname{dc}\left(z,k^{\prime}\right)$	$\operatorname{sc}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sn}\left(z,k^{\prime}\right)$
$\operatorname{cd}\left(iz,k\right)$ $\;=\;$	$\operatorname{nd}\left(z,k^{\prime}\right)$	$\operatorname{ns}\left(iz,k\right)$ $\;=\;$	$-i\operatorname{cs}\left(z,k^{\prime}\right)$
…
$\operatorname{nd}\left(iz,k\right)$ $\;=\;$	$\operatorname{cd}\left(z,k^{\prime}\right)$	$\operatorname{cs}\left(iz,k\right)$ $\;=\;$	$-i\operatorname{ns}\left(z,k^{\prime}\right)$

► …

4: 22.13 Derivatives and Differential Equations

… ►

Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.

► ►►►►►►

$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{sn}z)$ $\;=\;$	$\operatorname{cn}z\operatorname{dn}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{dc}z)$ =	${k^{\prime}}^{2}\operatorname{sc}z\operatorname{nc}z$
$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{cn}z)$ $\;=\;$	$-\operatorname{sn}z\operatorname{dn}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{nc}z)$ =	$\operatorname{sc}z\operatorname{dc}z$
$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{dn}z)$ $\;=\;$	$-k^{2}\operatorname{sn}z\operatorname{cn}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{sc}z)$ =	$\operatorname{dc}z\operatorname{nc}z$
…
$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{sd}z)$ $\;=\;$	$\operatorname{cd}z\operatorname{nd}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{ds}z)$ =	$-\operatorname{cs}z\operatorname{ns}z$
$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{nd}z)$ $\;=\;$	$k^{2}\operatorname{sd}z\operatorname{cd}z$	$\frac{\mathrm{d}}{\mathrm{d}z}(\operatorname{cs}z)$ =	$-\operatorname{ns}z\operatorname{ds}z$

► …

5: 22.4 Periods, Poles, and Zeros

… ►

Table 22.4.3: Half- or quarter-period shifts of variable for the Jacobian elliptic functions.

► ►►►►►►►

	$u$
…
$\operatorname{sn}u$	$\operatorname{cd}z$	$k^{-1}\operatorname{dc}z$	$k^{-1}\operatorname{ns}z$	$-\operatorname{sn}z$	$-\operatorname{sn}z$	$\operatorname{sn}z$
…
$\operatorname{dn}u$	$k^{\prime}\operatorname{nd}z$	$ik^{\prime}\operatorname{sc}z$	$-i\operatorname{cs}z$	$\operatorname{dn}z$	$-\operatorname{dn}z$	$-\operatorname{dn}z$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
…
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
…
$\operatorname{ns}u$	$\operatorname{dc}z$	$k\operatorname{cd}z$	$k\operatorname{sn}z$	$-\operatorname{ns}z$	$-\operatorname{ns}z$	$\operatorname{ns}z$
…

►

6: 22.21 Tables

§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. …

7: 22.7 Landen Transformations

… ►

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)},

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.2
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

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

►

22.7.4

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

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.4
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E4
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

… ►

22.7.8

\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)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable
A&S Ref:: 16.14.4
Permalink:: http://dlmf.nist.gov/22.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(ii), §22.7 and Ch.22

…

8: 22.14 Integrals

… ►See §22.16(i) for

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

. … ►

22.14.11

\int\operatorname{ds}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),

ⓘ

Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : 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, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
A&S Ref:: 16.24.11
Permalink:: http://dlmf.nist.gov/22.14.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

… ►

22.14.13

\int\frac{\,\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\ln\left(\frac{% \operatorname{sn}\left(x,k\right)}{\operatorname{cn}\left(x,k\right)+% \operatorname{dn}\left(x,k\right)}\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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
Referenced by:: §22.14(iii)
Permalink:: http://dlmf.nist.gov/22.14.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(iii), §22.14 and Ch.22

►

22.14.14

\int\frac{\operatorname{cn}\left(x,k\right)\,\mathrm{d}x}{\operatorname{sn}% \left(x,k\right)}=\frac{1}{2}\ln\left(\frac{1-\operatorname{dn}\left(x,k\right% )}{1+\operatorname{dn}\left(x,k\right)}\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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.14.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(iii), §22.14 and Ch.22

►

22.14.15

\int\frac{\operatorname{cn}\left(x,k\right)\,\mathrm{d}x}{{\operatorname{sn}}^% {2}\left(x,k\right)}=-\frac{\operatorname{dn}\left(x,k\right)}{\operatorname{% sn}\left(x,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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real and $k$ : modulus
Referenced by:: §22.14(iii)
Permalink:: http://dlmf.nist.gov/22.14.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(iii), §22.14 and Ch.22

…

9: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions

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

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

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

; the nine subsidiary Jacobian elliptic functions

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

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

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

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

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

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

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

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

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

; the amplitude function

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

; Jacobi’s epsilon and zeta functions

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

and

\mathrm{Z}\left(x|k\right)

. … ►The notation

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

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

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

is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for

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

are

\mathrm{sn}(z\mathpunct{|}m)

and

\mathrm{sn}(z,m)

with

m=k^{2}

; see Abramowitz and Stegun (1964) and Walker (1996). …

10: 22.17 Moduli Outside the Interval [0,1]

… ►

22.17.1

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

ⓘ

Symbols:: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

22.17.2

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

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.2
Permalink:: http://dlmf.nist.gov/22.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

►

22.17.3

\operatorname{cn}\left(z,1/k\right)=\operatorname{dn}\left(z/k,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 and $k$ : modulus
A&S Ref:: 16.11.3
Permalink:: http://dlmf.nist.gov/22.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

►

22.17.4

\operatorname{dn}\left(z,1/k\right)=\operatorname{cn}\left(z/k,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 and $k$ : modulus
A&S Ref:: 16.11.4
Permalink:: http://dlmf.nist.gov/22.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

22.17.7

\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k_{1}^{\prime},k_{% 1}\right),

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathrm{i}$ : imaginary unit, $z$ : complex, $k$ : modulus, $k_{1}$ : change of variable and $k_{1}^{\prime}$ : change of variable
A&S Ref:: 16.10.3 (modified)
Permalink:: http://dlmf.nist.gov/22.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

…

Jacobi elliptic

1—10 of 61 matching pages

1: 22.16 Related Functions

§22.16(i) Jacobi’s Amplitude ( am ) Function

Definition

Quasi-Periodicity

Integral Representation

Special Values

2: 22.8 Addition Theorems

3: 22.6 Elementary Identities