Jacobian elliptic functions

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1: 22.15 Inverse Functions

§22.15 Inverse Functions

►

§22.15(i) Definitions

… ►The principal values satisfy … ►

§22.15(ii) Representations as Elliptic Integrals

… ►

2: 22.2 Definitions

§22.2 Definitions

… ►

22.2.4

\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : 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:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►

22.2.9

\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.

ⓘ

Defines:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : 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:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►As a function of

z

, with fixed

k

, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … …

3: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

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§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).

4: 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)

. … ►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). …

5: 22.18 Mathematical Applications

§22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

… ►

Lemniscate

… ► … ►

6: 22.6 Elementary Identities

§22.6 Elementary Identities

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§22.6(ii) Double Argument

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§22.6(iii) Half Argument

… ►

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

… ►See §22.17.

7: 22.4 Periods, Poles, and Zeros

… ►

§22.4(i) Distribution

… ► … ►

…

Figure 22.4.1:

z

-plane. …

… ► ►

§22.4(ii) Graphical Interpretation via Glaisher’s Notation

…

8: 22.8 Addition Theorems

§22.8 Addition Theorems

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22.8.1

\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}v\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}u\operatorname{dn}u}{1-k^{2}{% \operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v},

ⓘ

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, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.1
Referenced by:: §22.18(iv)
Permalink:: http://dlmf.nist.gov/22.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

►

22.8.2

\operatorname{cn}(u+v)=\frac{\operatorname{cn}u\operatorname{cn}v-% \operatorname{sn}u\operatorname{dn}u\operatorname{sn}v\operatorname{dn}v}{1-k^% {2}{\operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v},

ⓘ

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, $u$ : complex, $v$ : complex and $k$ : modulus
A&S Ref:: 16.17.2
Referenced by:: §22.6(ii)
Permalink:: http://dlmf.nist.gov/22.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(i), §22.8 and Ch.22

… ►

§22.8(iii) Special Relations Between Arguments

… ►

9: 22.14 Integrals

… ►

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

… ►

22.14.1

\int\operatorname{sn}\left(x,k\right)\,\mathrm{d}x=k^{-1}\ln\left(% \operatorname{dn}\left(x,k\right)-k\operatorname{cn}\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
A&S Ref:: 16.24.1
Permalink:: http://dlmf.nist.gov/22.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.14(i), §22.14 and Ch.22

… ►

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