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22 Jacobian Elliptic Functions Properties 22.6 Elementary Identities 22.8 Addition Theorems

§22.7 Landen Transformations

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Permalink:: http://dlmf.nist.gov/22.7
See also:: Annotations for Ch.22

Contents

§22.7(i) Descending Landen Transformation
§22.7(ii) Ascending Landen Transformation
§22.7(iii) Generalized Landen Transformations

§22.7(i) Descending Landen Transformation

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Keywords:: Jacobian elliptic functions, Landen transformations, descending
Notes:: See Lawden (1989, pp. 79–80), Walker (1996, p. 148), Whittaker and Watson (1927, pp. 507–508).
Referenced by:: §22.17(ii), §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.i
See also:: Annotations for §22.7 and Ch.22

With

22.7.1		$k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},$
ⓘ Defines: $k_{1}$ : change of variable (locally) Symbols: $k^{\prime}$ : complementary modulus A&S Ref: 16.12.1 Referenced by: §22.20(iii) Permalink: http://dlmf.nist.gov/22.7.E1 Encodings: TeX, pMML, png See also: Annotations for §22.7(i), §22.7 and Ch.22

22.7.2		$\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable A&S Ref: 16.12.2 Referenced by: §22.20(iii) Permalink: http://dlmf.nist.gov/22.7.E2 Encodings: TeX, pMML, png See also: Annotations for §22.7(i), §22.7 and Ch.22

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: TeX, pMML, png See also: Annotations for §22.7(i), §22.7 and Ch.22

22.7.4		$\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}}^{2}\left(z/(1+k_{1}),k_% {1}\right)}.$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable A&S Ref: 16.12.4 Referenced by: §22.20(iii) Permalink: http://dlmf.nist.gov/22.7.E4 Encodings: TeX, pMML, png See also: Annotations for §22.7(i), §22.7 and Ch.22

§22.7(ii) Ascending Landen Transformation

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Keywords:: Jacobian elliptic functions, Landen transformations, ascending
Notes:: See Lawden (1989, pp. 79–80), Walker (1996, p. 148), Whittaker and Watson (1927, pp. 507–508).
Referenced by:: §22.17(ii), §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.ii
See also:: Annotations for §22.7 and Ch.22

With

22.7.5		$\displaystyle k_{2}$	$\displaystyle=\frac{2\sqrt{k}}{1+k},$
22.7.5		$\displaystyle k_{2}^{\prime}$	$\displaystyle=\frac{1-k}{1+k},$
ⓘ Defines: $k_{2}$ : change of variable (locally) and $k_{2}^{\prime}$ : change of variable (locally) Symbols: $k$ : modulus A&S Ref: 16.14.1 Permalink: http://dlmf.nist.gov/22.7.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §22.7(ii), §22.7 and Ch.22

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: TeX, pMML, png See also: Annotations for §22.7(ii), §22.7 and Ch.22

22.7.7		$\operatorname{cn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)-k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)},$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable A&S Ref: 16.14.3 Permalink: http://dlmf.nist.gov/22.7.E7 Encodings: TeX, pMML, png See also: Annotations for §22.7(ii), §22.7 and Ch.22

22.7.8		$\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.$
ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus, $k_{2}$ : change of variable and $k_{2}^{\prime}$ : change of variable A&S Ref: 16.14.4 Permalink: http://dlmf.nist.gov/22.7.E8 Encodings: TeX, pMML, png See also: Annotations for §22.7(ii), §22.7 and Ch.22

§22.7(iii) Generalized Landen Transformations

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Keywords:: Jacobian elliptic functions, Landen transformations, generalized
Permalink:: http://dlmf.nist.gov/22.7.iii
See also:: Annotations for §22.7 and Ch.22

See Khare and Sukhatme (2004).

© 2010–2020 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.1.0; Release date 2020-12-15. A printed companion is available. 22.6 Elementary Identities 22.8 Addition Theorems