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

§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 $k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},$

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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 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},$

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, : complex, : 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 $\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}\right)},$

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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, : complex, : 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 $\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}}^{2}\left(z/(1+k_{1}),k_% {1}\right)}.$

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Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, : complex, : 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

$k_{2}=\frac{2\sqrt{k}}{1+k},$

$k_{2}^{\prime}=\frac{1-k}{1+k},$

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Defines:: $k_{2}$ : change of variable (locally) and $k_{2}^{\prime}$ : change of variable (locally)
Symbols:: : 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 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},$

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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, : complex, : 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 $\operatorname{cn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)-k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)},$

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, : complex, : 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 $\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.$

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Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, : complex, : 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).

22.6 Elementary Identities 22.8 Addition Theorems

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§22.7 Landen Transformations

Contents

§22.7(i) Descending Landen Transformation

§22.7(ii) Ascending Landen Transformation

§22.7(iii) Generalized Landen Transformations