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1: Gloria Wiersma

… ► 1937 in Washington, DC) joined the NIST staff in 1973, where she occupied various positions providing support for the Physics Laboratory until 1993. …

2: 22.13 Derivatives and Differential Equations

… ►

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

► ►►►►

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

► … ►

22.13.7

\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{% 2}=\left({\operatorname{dc}}^{2}\left(z,k\right)-1\right)\left({\operatorname{% dc}}^{2}\left(z,k\right)-k^{2}\right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

… ►

22.13.19

\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\operatorname{dc}\left(z,k\right)=-(% 1+k^{2})\operatorname{dc}\left(z,k\right)+2{\operatorname{dc}}^{3}\left(z,k% \right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.13.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(iii), §22.13 and Ch.22

…

3: 22.4 Periods, Poles, and Zeros

… ►

Table 22.4.1: Periods and poles of Jacobian elliptic functions.

► ►►►

Periods	$z$ -Poles
Periods	…
$4K$ , $2iK^{\prime}$	$\operatorname{sn}$	$\operatorname{cd}$	$\operatorname{dc}$	$\operatorname{ns}$
…

► … ►

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

► ►►►►►►

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

►

4: 22.6 Elementary Identities

… ►

22.6.3

{k^{\prime}}^{2}{\operatorname{sc}}^{2}\left(z,k\right)+1={\operatorname{dc}}^% {2}\left(z,k\right)={k^{\prime}}^{2}{\operatorname{nc}}^{2}\left(z,k\right)+k^% {2},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.9.3
Permalink:: http://dlmf.nist.gov/22.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(i), §22.6 and Ch.22

… ►

22.6.11

\operatorname{dc}\left(2z,k\right)=\frac{{\operatorname{dc}}^{2}\left(z,k% \right)+{k^{\prime}}^{2}{\operatorname{sc}}^{2}\left(z,k\right){\operatorname{% nc}}^{2}\left(z,k\right)}{1-{k^{\prime}}^{2}{\operatorname{sc}}^{4}\left(z,k% \right)},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.12

\operatorname{nc}\left(2z,k\right)=\frac{{\operatorname{nc}}^{2}\left(z,k% \right)+{\operatorname{sc}}^{2}\left(z,k\right){\operatorname{dc}}^{2}\left(z,% k\right)}{1-{k^{\prime}}^{2}{\operatorname{sc}}^{4}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

►

22.6.13

\operatorname{sc}\left(2z,k\right)=\frac{2\operatorname{sc}\left(z,k\right)% \operatorname{dc}\left(z,k\right)\operatorname{nc}\left(z,k\right)}{1-{k^{% \prime}}^{2}{\operatorname{sc}}^{4}\left(z,k\right)},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/22.6.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §22.6(ii), §22.6 and Ch.22

… ►

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.

► ►►

$\operatorname{sn}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sc}\left(z,k^{\prime}\right)$	$\operatorname{dc}\left(iz,k\right)$ $\;=\;$	$\operatorname{dn}\left(z,k^{\prime}\right)$
…

► …

5: 22.8 Addition Theorems

… ►

22.8.7

\operatorname{dc}(u+v)=\frac{\operatorname{dc}u\operatorname{dc}v+{k^{\prime}}% ^{2}\operatorname{sc}u\operatorname{nc}u\operatorname{sc}v\operatorname{nc}v}{% 1-{k^{\prime}}^{2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

►

22.8.8

\operatorname{nc}(u+v)=\frac{\operatorname{nc}u\operatorname{nc}v+% \operatorname{sc}u\operatorname{dc}u\operatorname{sc}v\operatorname{dc}v}{1-{k% ^{\prime}}^{2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

►

22.8.9

\operatorname{sc}(u+v)=\frac{\operatorname{sc}u\operatorname{dc}v\operatorname% {nc}v+\operatorname{sc}v\operatorname{dc}u\operatorname{nc}u}{1-{k^{\prime}}^{% 2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $u$ : complex, $v$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

…

6: 22.14 Integrals

… ►

22.14.7

\int\operatorname{dc}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{nc}% \left(x,k\right)+\operatorname{sc}\left(x,k\right)\right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : 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:: 16.24.7
Permalink:: http://dlmf.nist.gov/22.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.8

\int\operatorname{nc}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}\ln\left(% \operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{sc}\left(x,k\right)% \right),

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.24.8
Permalink:: http://dlmf.nist.gov/22.14.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.9

\int\operatorname{sc}\left(x,k\right)\,\mathrm{d}x={k^{\prime}}^{-1}\ln\left(% \operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{nc}\left(x,k\right)% \right).

ⓘ

Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real, $k$ : modulus and $k^{\prime}$ : complementary modulus
A&S Ref:: 16.24.9
Permalink:: http://dlmf.nist.gov/22.14.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

…

7: 22.3 Graphics

… ►Line graphs of the functions

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

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

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

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

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

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

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

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

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

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

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

, and

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

for representative values of real

x

and real

k

illustrating the near trigonometric (

k=0

), and near hyperbolic (

k=1

) limits. … ►

See accompanying text — Figure 22.3.20: $\operatorname{dc}\left(x+iy,k\right)$ for $k=0.99$ , $-3K\leq x\leq 3K$ , $0\leq y\leq 4K^{\prime}$ . … Magnify 3D Help

…

8: 22.5 Special Values

… ►

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

► ►►►

	$z$
…
$\operatorname{dc}z$	$1,0$	$\infty,-1$	$0,k$	$k,0$	$-1,0$	$-1,0$	$1,0$
…

► … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as

k\to 0

► ►►

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

► ►

Table 22.5.4: Limiting forms of Jacobian elliptic functions as

k\to 1

► ►►

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\tanh z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\coth z$
…

► …

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)

. …

10: 22.15 Inverse Functions

… ►are denoted respectively by … ►

22.15.18

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

1\leq x<\infty

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcdc}\left(\NVar{x},\NVar{k}\right)$ : inverse Jacobian elliptic function, $x$ : real and $k$ : modulus
A&S Ref:: 17.4.47
Permalink:: http://dlmf.nist.gov/22.15.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §22.15(ii), §22.15 and Ch.22

…