22 Jacobian Elliptic Functions Properties 22.1 Special Notation 22.3 Graphics

§22.2 Definitions

ⓘ

Keywords:: Jacobian elliptic functions, Neville’s theta functions, analytic properties, definitions, modulus, nome, periods, relations to Jacobian elliptic functions, relations to other functions, theta functions
Notes:: See Lawden (1989, §2.1), Walker (1996, §§5.1, 6.2), Walker (2003), Whittaker and Watson (1927, §22.1).
Referenced by:: §19.25(v), §20.1, §20.11(iii), §20.7(ii), §20.9(ii), §22.11, §22.20(i), §22.21, §29.2(i), §29.3(v), §29.6(i), §29.8, §31.2(iv), Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/22.2
Clarification (effective with 1.0.7):: The part of the final sentence of this section that follows (22.2.12) has been clarified.
Suggested 2013-11-15 by Howard Cohl
See also:: Annotations for Ch.22

The nome $q$ is given in terms of the modulus $k$ by

22.2.1		$q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right),$
ⓘ Defines: $q$ : nome Symbols: $\pi$ : 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, $\exp\NVar{z}$ : exponential function and $k$ : modulus Referenced by: §20.9(i), §22.16(i) Permalink: http://dlmf.nist.gov/22.2.E1 Encodings: TeX, pMML, png See also: Annotations for §22.2 and Ch.22

where $K\left(k\right)$ , ${K^{\prime}}\left(k\right)$ are defined in §19.2(ii). Inversely,

22.2.2	$\displaystyle k$	$\displaystyle=\frac{{\theta_{2}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,% q\right)},$
	$\displaystyle k^{\prime}$	$\displaystyle=\frac{{\theta_{4}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,% q\right)},$
	$\displaystyle K\left(k\right)$	$\displaystyle=\frac{\pi}{2}{\theta_{3}}^{2}\left(0,q\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $q$ : nome, $k$ : modulus and $k^{\prime}$ : complementary modulus Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/22.2.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §22.2 and Ch.22

where $k^{\prime}=\sqrt{1-k^{2}}$ and the theta functions are defined in §20.2(i).

With

22.2.3		$\zeta=\frac{\pi z}{2K\left(k\right)},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable Permalink: http://dlmf.nist.gov/22.2.E3 Encodings: TeX, pMML, png See also: Annotations for §22.2 and Ch.22

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: TeX, pMML, png See also: Annotations for §22.2 and Ch.22

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: TeX, pMML, png 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: TeX, pMML, png See also: Annotations for §22.2 and Ch.22

22.2.7		$\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3}}^{2}\left(0,q\right)}{% \theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(% \zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}% \left(z,k\right)},$
ⓘ Defines: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : 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.E7 Encodings: TeX, pMML, png 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: TeX, pMML, png 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: TeX, pMML, png 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. Each is meromorphic in $z$ for fixed $k$ , with simple poles and simple zeros, and each is meromorphic in $k$ for fixed $z$ . For $k\in[0,1]$ , all functions are real for $z\in\mathbb{R}$ .

Glaisher’s Notation

ⓘ

Keywords:: Neville’s, theta functions
See also:: Annotations for §22.2 and Ch.22

The Jacobian functions are related in the following way. Let $\mathrm{p}$ , $\mathrm{q}$ , $\mathrm{r}$ be any three of the letters $\mathrm{s}$ , $\mathrm{c}$ , $\mathrm{d}$ , $\mathrm{n}$ . Then

22.2.10		$\operatorname{pq}\left(z,k\right)=\frac{\operatorname{pr}\left(z,k\right)}{% \operatorname{qr}\left(z,k\right)}=\frac{1}{\operatorname{qp}\left(z,k\right)},$
ⓘ Defines: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function Symbols: $z$ : complex and $k$ : modulus A&S Ref: 16.3.4 Referenced by: §22.14(iv), §22.20(ii), §22.4(ii), §22.5(i) Permalink: http://dlmf.nist.gov/22.2.E10 Encodings: TeX, pMML, png See also: Annotations for §22.2, §22.2 and Ch.22

with the convention that functions with the same two letters are replaced by unity; e.g. $\operatorname{ss}\left(z,k\right)=1$ .

The six functions containing the letter $\mathrm{s}$ in their two-letter name are odd in $z$ ; the other six are even in $z$ .

In terms of Neville’s theta functions (§20.1)

22.2.11		$\operatorname{pq}\left(z,k\right)=\ifrac{\theta_{p}\left(z\middle\|\tau\right)}% {\theta_{q}\left(z\middle\|\tau\right)},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $q$ : nome, $z$ : complex, $k$ : modulus and $\tau$ : ratio A&S Ref: 16.36.3 Referenced by: §22.2 Permalink: http://dlmf.nist.gov/22.2.E11 Encodings: TeX, pMML, png See also: Annotations for §22.2, §22.2 and Ch.22

where

22.2.12		$\tau=\ifrac{\mathrm{i}{K^{\prime}}\left(k\right)}{K\left(k\right)},$
ⓘ Symbols: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $k$ : modulus and $\tau$ : ratio Referenced by: §22.2 Permalink: http://dlmf.nist.gov/22.2.E12 Encodings: TeX, pMML, png See also: Annotations for §22.2, §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 $1,2,3,4$ .

22.1 Special Notation 22.3 Graphics

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§22.2 Definitions

Glaisher’s Notation

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!