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22 Jacobian Elliptic Functions Properties 22.5 Special Values 22.7 Landen Transformations

§22.6 Elementary Identities

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Keywords:: Jacobian elliptic functions, elementary identities
Permalink:: http://dlmf.nist.gov/22.6
See also:: Annotations for Ch.22

Contents

§22.6(i) Sums of Squares

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Keywords:: Jacobian elliptic functions, sums of squares
Notes:: See Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 491–494).
Permalink:: http://dlmf.nist.gov/22.6.i
See also:: Annotations for §22.6 and Ch.22

22.6.1		${\operatorname{sn}}^{2}\left(z,k\right)+{\operatorname{cn}}^{2}\left(z,k\right% )=k^{2}{\operatorname{sn}}^{2}\left(z,k\right)+{\operatorname{dn}}^{2}\left(z,% k\right)=1,$
ⓘ 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 and $k$ : modulus A&S Ref: 16.9.1 Referenced by: §22.11, §22.11, §22.20(ii), §22.6(ii) Permalink: http://dlmf.nist.gov/22.6.E1 Encodings: TeX, pMML, png See also: Annotations for §22.6(i), §22.6 and Ch.22

22.6.2		$1+{\operatorname{cs}}^{2}\left(z,k\right)=k^{2}+{\operatorname{ds}}^{2}\left(z% ,k\right)={\operatorname{ns}}^{2}\left(z,k\right),$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus A&S Ref: 16.9.4 Permalink: http://dlmf.nist.gov/22.6.E2 Encodings: TeX, pMML, png See also: Annotations for §22.6(i), §22.6 and Ch.22

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: TeX, pMML, png See also: Annotations for §22.6(i), §22.6 and Ch.22

22.6.4		$-k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{2}\left(z,k\right)=k^{2}({% \operatorname{cd}}^{2}\left(z,k\right)-1)={k^{\prime}}^{2}(1-{\operatorname{nd% }}^{2}\left(z,k\right)).$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.9.2 Permalink: http://dlmf.nist.gov/22.6.E4 Encodings: TeX, pMML, png See also: Annotations for §22.6(i), §22.6 and Ch.22

§22.6(ii) Double Argument

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Keywords:: Jacobian elliptic functions, double argument
Notes:: See Lawden (1989, pp. 33–36). For (22.6.6) and (22.6.7) set $u=v=z$ in (22.8.2) and use (22.6.1) repeatedly.
Permalink:: http://dlmf.nist.gov/22.6.ii
See also:: Annotations for §22.6 and Ch.22

22.6.5		$\operatorname{sn}\left(2z,k\right)=\frac{2\operatorname{sn}\left(z,k\right)% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)}{1-k^{2}{% \operatorname{sn}}^{4}\left(z,k\right)},$
ⓘ 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 and $k$ : modulus A&S Ref: 16.18.1 Permalink: http://dlmf.nist.gov/22.6.E5 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22

22.6.6		$\operatorname{cn}\left(2z,k\right)=\frac{{\operatorname{cn}}^{2}\left(z,k% \right)-{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,% k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{\operatorname% {cn}}^{4}\left(z,k\right)-{k^{\prime}}^{2}{\operatorname{sn}}^{4}\left(z,k% \right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)},$
ⓘ 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 $k^{\prime}$ : complementary modulus A&S Ref: 16.18.2 Referenced by: §22.6(ii) Permalink: http://dlmf.nist.gov/22.6.E6 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22

22.6.7		$\operatorname{dn}\left(2z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z,k% \right)-k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}% \left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{% \operatorname{dn}}^{4}\left(z,k\right)+k^{2}{k^{\prime}}^{2}{\operatorname{sn}% }^{4}\left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}.$
ⓘ 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 $k^{\prime}$ : complementary modulus A&S Ref: 16.18.3 Referenced by: §22.6(ii), Erratum (V1.0.7) for Equation (22.6.7) Permalink: http://dlmf.nist.gov/22.6.E7 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the term $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}\left(z,k\right)$ was given incorrectly as $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,k\right)$ . Reported 2014-02-28 by Svante Janson See also: Annotations for §22.6(ii), §22.6 and Ch.22

22.6.8		$\displaystyle\operatorname{cd}\left(2z,k\right)$	$\displaystyle=\frac{{\operatorname{cd}}^{2}\left(z,k\right)-{k^{\prime}}^{2}{% \operatorname{sd}}^{2}\left(z,k\right){\operatorname{nd}}^{2}\left(z,k\right)}% {1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k\right)},$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.6.E8 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.9		$\displaystyle\operatorname{sd}\left(2z,k\right)$	$\displaystyle=\frac{2\operatorname{sd}\left(z,k\right)\operatorname{cd}\left(z% ,k\right)\operatorname{nd}\left(z,k\right)}{1+k^{2}{k^{\prime}}^{2}{% \operatorname{sd}}^{4}\left(z,k\right)},$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.6.E9 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.10		$\displaystyle\operatorname{nd}\left(2z,k\right)$	$\displaystyle=\frac{{\operatorname{nd}}^{2}\left(z,k\right)+k^{2}{% \operatorname{sd}}^{2}\left(z,k\right){\operatorname{cd}}^{2}\left(z,k\right)}% {1+k^{2}{k^{\prime}}^{2}{\operatorname{sd}}^{4}\left(z,k\right)},$
ⓘ Symbols: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.6.E10 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.11		$\displaystyle\operatorname{dc}\left(2z,k\right)$	$\displaystyle=\frac{{\operatorname{dc}}^{2}\left(z,k\right)+{k^{\prime}}^{2}{% \operatorname{sc}}^{2}\left(z,k\right){\operatorname{nc}}^{2}\left(z,k\right)}% {1-{k^{\prime}}^{2}{\operatorname{sc}}^{4}\left(z,k\right)},$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22

22.6.12		$\displaystyle\operatorname{nc}\left(2z,k\right)$	$\displaystyle=\frac{{\operatorname{nc}}^{2}\left(z,k\right)+{\operatorname{sc}% }^{2}\left(z,k\right){\operatorname{dc}}^{2}\left(z,k\right)}{1-{k^{\prime}}^{% 2}{\operatorname{sc}}^{4}\left(z,k\right)},$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.13		$\displaystyle\operatorname{sc}\left(2z,k\right)$	$\displaystyle=\frac{2\operatorname{sc}\left(z,k\right)\operatorname{dc}\left(z% ,k\right)\operatorname{nc}\left(z,k\right)}{1-{k^{\prime}}^{2}{\operatorname{% sc}}^{4}\left(z,k\right)},$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.14		$\displaystyle\operatorname{ns}\left(2z,k\right)$	$\displaystyle=\frac{{\operatorname{ns}}^{4}\left(z,k\right)-k^{2}}{2% \operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k\right)% \operatorname{ns}\left(z,k\right)},$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus Permalink: http://dlmf.nist.gov/22.6.E14 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.15		$\displaystyle\operatorname{ds}\left(2z,k\right)$	$\displaystyle=\frac{k^{2}{k^{\prime}}^{2}+{\operatorname{ds}}^{4}\left(z,k% \right)}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k\right)% \operatorname{ns}\left(z,k\right)},$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.6.E15 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.16		$\displaystyle\operatorname{cs}\left(2z,k\right)$	$\displaystyle=\frac{{\operatorname{cs}}^{4}\left(z,k\right)-{k^{\prime}}^{2}}{% 2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k\right)% \operatorname{ns}\left(z,k\right)}.$
ⓘ Symbols: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/22.6.E16 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22

See also Carlson (2004).

22.6.17		$\displaystyle\frac{1-\operatorname{cn}\left(2z,k\right)}{1+\operatorname{cn}% \left(2z,k\right)}$	$\displaystyle=\frac{{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}% ^{2}\left(z,k\right)}{{\operatorname{cn}}^{2}\left(z,k\right)},$
ⓘ 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 and $k$ : modulus A&S Ref: 16.18.4 Permalink: http://dlmf.nist.gov/22.6.E17 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22
22.6.18		$\displaystyle\frac{1-\operatorname{dn}\left(2z,k\right)}{1+\operatorname{dn}% \left(2z,k\right)}$	$\displaystyle=\frac{k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname% {cn}}^{2}\left(z,k\right)}{{\operatorname{dn}}^{2}\left(z,k\right)}.$
ⓘ 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 and $k$ : modulus A&S Ref: 16.18.5 Permalink: http://dlmf.nist.gov/22.6.E18 Encodings: TeX, pMML, png See also: Annotations for §22.6(ii), §22.6 and Ch.22

§22.6(iii) Half Argument

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Keywords:: Jacobian elliptic functions, half argument
Notes:: See Lawden (1989, pp. 33–36).
Permalink:: http://dlmf.nist.gov/22.6.iii
See also:: Annotations for §22.6 and Ch.22

22.6.19		$\displaystyle{\operatorname{sn}}^{2}\left(\tfrac{1}{2}z,k\right)$	$\displaystyle=\frac{1-\operatorname{cn}\left(z,k\right)}{1+\operatorname{dn}% \left(z,k\right)}=\frac{1-\operatorname{dn}\left(z,k\right)}{k^{2}(1+% \operatorname{cn}\left(z,k\right))}=\frac{\operatorname{dn}\left(z,k\right)-k^% {2}\operatorname{cn}\left(z,k\right)-{k^{\prime}}^{2}}{k^{2}(\operatorname{dn}% \left(z,k\right)-\operatorname{cn}\left(z,k\right))},$
ⓘ 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 $k^{\prime}$ : complementary modulus A&S Ref: 16.19.1 Permalink: http://dlmf.nist.gov/22.6.E19 Encodings: TeX, pMML, png See also: Annotations for §22.6(iii), §22.6 and Ch.22
22.6.20		$\displaystyle{\operatorname{cn}}^{2}\left(\tfrac{1}{2}z,k\right)$	$\displaystyle=\frac{-{k^{\prime}}^{2}+\operatorname{dn}\left(z,k\right)+k^{2}% \operatorname{cn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}\left(z,k\right))}% =\frac{{k^{\prime}}^{2}(1-\operatorname{dn}\left(z,k\right))}{k^{2}(% \operatorname{dn}\left(z,k\right)-\operatorname{cn}\left(z,k\right))}=\frac{{k% ^{\prime}}^{2}(1+\operatorname{cn}\left(z,k\right))}{{k^{\prime}}^{2}+% \operatorname{dn}\left(z,k\right)-k^{2}\operatorname{cn}\left(z,k\right)},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.19.2 Permalink: http://dlmf.nist.gov/22.6.E20 Encodings: TeX, pMML, png See also: Annotations for §22.6(iii), §22.6 and Ch.22
22.6.21		$\displaystyle{\operatorname{dn}}^{2}\left(\tfrac{1}{2}z,k\right)$	$\displaystyle=\frac{k^{2}\operatorname{cn}\left(z,k\right)+\operatorname{dn}% \left(z,k\right)+{k^{\prime}}^{2}}{1+\operatorname{dn}\left(z,k\right)}=\frac{% {k^{\prime}}^{2}(1-\operatorname{cn}\left(z,k\right))}{\operatorname{dn}\left(% z,k\right)-\operatorname{cn}\left(z,k\right)}=\frac{{k^{\prime}}^{2}(1+% \operatorname{dn}\left(z,k\right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,% k\right)-k^{2}\operatorname{cn}\left(z,k\right)}.$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.19.3 Permalink: http://dlmf.nist.gov/22.6.E21 Encodings: TeX, pMML, png See also: Annotations for §22.6(iii), §22.6 and Ch.22

If $\{\mbox{p,q,r}\}$ is any permutation of $\{\mbox{c,d,n}\}$ , then

22.6.22		${\operatorname{pq}}^{2}\left(\tfrac{1}{2}z,k\right)=\frac{\operatorname{ps}% \left(z,k\right)+\operatorname{rs}\left(z,k\right)}{\operatorname{qs}\left(z,k% \right)+\operatorname{rs}\left(z,k\right)}=\frac{\operatorname{pq}\left(z,k% \right)+\operatorname{rq}\left(z,k\right)}{1+\operatorname{rq}\left(z,k\right)% }=\frac{\operatorname{pr}\left(z,k\right)+1}{\operatorname{qr}\left(z,k\right)% +1}.$
ⓘ Symbols: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $z$ : complex and $k$ : modulus Referenced by: §22.6(iii) Permalink: http://dlmf.nist.gov/22.6.E22 Encodings: TeX, pMML, png See also: Annotations for §22.6(iii), §22.6 and Ch.22

For (22.6.22) and similar results, see Carlson (2004).

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

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Keywords:: Jacobian elliptic functions, Jacobi’s imaginary transformation, rotation of argument
Notes:: See Lawden (1989, pp. 38–40), Walker (1996, pp. 126–131), Whittaker and Watson (1927, pp. 505–508).
Permalink:: http://dlmf.nist.gov/22.6.iv
See also:: Annotations for §22.6 and Ch.22

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

$\operatorname{sn}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sc}\left(z,k^{\prime}\right)$	$\operatorname{dc}\left(iz,k\right)$ $\;=\;$	$\operatorname{dn}\left(z,k^{\prime}\right)$
$\operatorname{cn}\left(iz,k\right)$ $\;=\;$	$\operatorname{nc}\left(z,k^{\prime}\right)$	$\operatorname{nc}\left(iz,k\right)$ $\;=\;$	$\operatorname{cn}\left(z,k^{\prime}\right)$
$\operatorname{dn}\left(iz,k\right)$ $\;=\;$	$\operatorname{dc}\left(z,k^{\prime}\right)$	$\operatorname{sc}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sn}\left(z,k^{\prime}\right)$
$\operatorname{cd}\left(iz,k\right)$ $\;=\;$	$\operatorname{nd}\left(z,k^{\prime}\right)$	$\operatorname{ns}\left(iz,k\right)$ $\;=\;$	$-i\operatorname{cs}\left(z,k^{\prime}\right)$
$\operatorname{sd}\left(iz,k\right)$ $\;=\;$	$i\operatorname{sd}\left(z,k^{\prime}\right)$	$\operatorname{ds}\left(iz,k\right)$ $\;=\;$	$-i\operatorname{ds}\left(z,k^{\prime}\right)$
$\operatorname{nd}\left(iz,k\right)$ $\;=\;$	$\operatorname{cd}\left(z,k^{\prime}\right)$	$\operatorname{cs}\left(iz,k\right)$ $\;=\;$	$-i\operatorname{ns}\left(z,k^{\prime}\right)$

§22.6(v) Change of Modulus

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Keywords:: Jacobian elliptic functions, elementary identities
Permalink:: http://dlmf.nist.gov/22.6.v
See also:: Annotations for §22.6 and Ch.22

See §22.17.

22.5 Special Values 22.7 Landen Transformations

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