	$u$
…
$\operatorname{sn}u$	$\operatorname{cd}z$	$k^{-1}\operatorname{dc}z$	$k^{-1}\operatorname{ns}z$	$-\operatorname{sn}z$	$-\operatorname{sn}z$	$\operatorname{sn}z$
…
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
…

►

3: 22.8 Addition Theorems

… ►

22.8.4

\operatorname{cd}(u+v)=\frac{\operatorname{cd}u\operatorname{cd}v-{k^{\prime}}% ^{2}\operatorname{sd}u\operatorname{nd}u\operatorname{sd}v\operatorname{nd}v}{% 1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

►

22.8.5

\operatorname{sd}(u+v)=\frac{\operatorname{sd}u\operatorname{cd}v\operatorname% {nd}v+\operatorname{sd}v\operatorname{cd}u\operatorname{nd}u}{1+k^{2}{k^{% \prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

►

22.8.6

\operatorname{nd}(u+v)=\frac{\operatorname{nd}u\operatorname{nd}v+k^{2}% \operatorname{sd}u\operatorname{cd}u\operatorname{sd}v\operatorname{cd}v}{1+k^% {2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

… ►

22.8.19

z_{1}+z_{2}+z_{3}+z_{4}=0.

ⓘ

Symbols:: $z$ : complex
Permalink:: http://dlmf.nist.gov/22.8.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(iii), §22.8 and Ch.22

… ►

22.8.23

\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.

ⓘ

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, $\det$ : determinant, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(iii), §22.8 and Ch.22

…

4: 22.16 Related Functions

… ►

22.16.18

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

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : 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
A&S Ref:: 16.26.4
Permalink:: http://dlmf.nist.gov/22.16.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

►

22.16.19

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

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : 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, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.26.5
Permalink:: http://dlmf.nist.gov/22.16.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

►

22.16.20

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

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : 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, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.26.6
Permalink:: http://dlmf.nist.gov/22.16.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

… ►

22.16.28

\mathcal{E}\left(x+K,k\right)=\mathcal{E}\left(x,k\right)+E\left(k\right)-k^{2% }\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right),

ⓘ

Symbols:: $\mathcal{E}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s epsilon function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

… ►

22.16.33

\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k\right)-k^{2}\operatorname{sn}% \left(x,k\right)\operatorname{cd}\left(x,k\right),

ⓘ

Symbols:: $\mathrm{Z}\left(\NVar{x}|\NVar{k}\right)$ : Jacobi’s zeta function, $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $x$ : real and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.16.E33
Encodings:: pMML, png, TeX
See also:: Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

…

5: 22.14 Integrals

… ►

22.14.4

\int\operatorname{cd}\left(x,k\right)\,\mathrm{d}x=k^{-1}\ln\left(% \operatorname{nd}\left(x,k\right)+k\operatorname{sd}\left(x,k\right)\right),

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : 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:: 16.24.4
Permalink:: http://dlmf.nist.gov/22.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.5

\int\operatorname{sd}\left(x,k\right)\,\mathrm{d}x=(kk^{\prime})^{-1}% \operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right),

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.24.5
Permalink:: http://dlmf.nist.gov/22.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.6

\int\operatorname{nd}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}% \operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)\right).

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Arccos}\NVar{z}$ : general arccosine function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.24.6
Permalink:: http://dlmf.nist.gov/22.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

… ►

22.14.16

\int_{0}^{K\left(k\right)}\ln\left(\operatorname{sn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)-\tfrac{1}{2}K\left(k% \right)\ln k,

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $k$ : modulus
Referenced by:: Erratum (V1.0.28) for Equations (22.14.16), (22.14.17)
Permalink:: http://dlmf.nist.gov/22.14.E16
Encodings:: pMML, png, TeX
Correction (effective with 1.0.28):: Originally, a factor of $\pi$ was missing from the term containing the $\tfrac{1}{4}{K^{\prime}}\left(k\right)$ .
Suggested 2020-08-06 by Fred Hucht
See also:: Annotations for §22.14(iv), §22.14 and Ch.22

►

22.14.17

\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\,% \mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)+\tfrac{1}{2}K\left(k% \right)\ln\left(k^{\prime}/k\right),

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : modulus and $k^{\prime}$ : complementary modulus
Referenced by:: Erratum (V1.0.28) for Equations (22.14.16), (22.14.17)
Permalink:: http://dlmf.nist.gov/22.14.E17
Encodings:: pMML, png, TeX
Correction (effective with 1.0.28):: Originally, a factor of $\pi$ was missing from the term containing the $\tfrac{1}{4}{K^{\prime}}\left(k\right)$ .
Suggested 2020-08-06 by Fred Hucht
See also:: Annotations for §22.14(iv), §22.14 and Ch.22

…

6: 22.2 Definitions

… ►

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

►

22.2.6

\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{3}% \left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nd}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : 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
Referenced by:: §22.20(ii)
Permalink:: http://dlmf.nist.gov/22.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►

22.2.8

\operatorname{cd}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q% \right)}=\frac{1}{\operatorname{dc}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : 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.E8
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

7: 22.15 Inverse Functions

… ►are denoted respectively by … ►

22.15.9

\eta=\pm\operatorname{arccn}\left(x,k\right)+4mK,

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{arccn}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real, $k$ : modulus, $\eta$ : solution and $m$ : integer
Permalink:: http://dlmf.nist.gov/22.15.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.15(i), §22.15 and Ch.22

… ►

22.15.15

\operatorname{arccd}\left(x,k\right)=\int_{x}^{1}\frac{\,\mathrm{d}t}{\sqrt{(1% -t^{2})(1-k^{2}t^{2})}},

-1\leq x\leq 1

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccd}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus
A&S Ref:: 17.4.46
Permalink:: http://dlmf.nist.gov/22.15.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §22.15(ii), §22.15 and Ch.22

…

8: 22.13 Derivatives and Differential Equations

… ►

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

► ►►►►►

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