22 Jacobian Elliptic Functions Properties 22.1 Special Notation 22.3 Graphics

§22.2 Definitions

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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

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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

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