elliptic cases of R-a(b;z)

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21: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

►

§22.17(i) Real or Purely Imaginary Moduli

… ►

§22.17(ii) Complex Moduli

►When

z

is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of

k^{2}

. …For proofs of these results and further information see Walker (2003).

22: 22.13 Derivatives and Differential Equations

… ►

§22.13(i) Derivatives

… ►

§22.13(ii) First-Order Differential Equations

… ►

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

… ►

22.13.10

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

ⓘ

Symbols:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : 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.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §22.13(ii), §22.13 and Ch.22

… ►

§22.13(iii) Second-Order Differential Equations

…

23: 22.1 Special Notation

… ► ►►►

$x, y$	real variables.
…
$K$ , ${K^{\prime}}$	$K\left(k\right)$ , ${K^{\prime}}\left(k\right)=K\left(k^{\prime}\right)$ (complete elliptic integrals of the first kind (§19.2(ii))).
…

… ►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). …

24: 19.6 Special Cases

§19.6 Special Cases

►

§19.6(i) Complete Elliptic Integrals

… ►Exact values of

K\left(k\right)

and

E\left(k\right)

for various special values of

k

are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006). ►

§19.6(ii) $F\left(\phi,k\right)$

… ►

§19.6(iii) $E\left(\phi,k\right)$

…

25: 22.14 Integrals

… ►

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

… ►See §22.16(i) for

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

. … ►

§22.14(ii) Indefinite Integrals of Powers of Jacobian Elliptic Functions

… ► ►

§22.14(iv) Definite Integrals

…

26: 22.10 Maclaurin Series

… ►

§22.10(i) Maclaurin Series in $z$

… ►The full expansions converge when

|z|<\min\left(K\left(k\right),{K^{\prime}}\left(k\right)\right)

. ►

§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.

27: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►There are four important special cases of (19.14.4)–(19.14.6), as follows. … ►(These four cases include 12 integrals in Abramowitz and Stegun (1964, p. 596).) ►

§19.14(ii) General Case

… ►

28: 22.7 Landen Transformations

… ►

§22.7(i) Descending Landen Transformation

… ►

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(ii) Ascending Landen Transformation

… ►

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(iii) Generalized Landen Transformations

…

29: 22.3 Graphics

… ►

§22.3(i) Real Variables: Line Graphs

►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. … ►

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

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