§22.7 Landen Transformations

Permalink:: http://dlmf.nist.gov/22.7

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

§22.7(i) Descending Landen Transformation

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

With

22.7.1 $k_{1}=\frac{1-k^{{\prime}}}{1+k^{{\prime}}},$

Symbols:: $k$ : modulus and $k^{{\prime}}$ : 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

22.7.2 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\frac{(1+k_{1})\mathop{\mathrm{sn}\/}\nolimits\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z/(1+k_{1}),k_{1}\right)},$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.12.2
Referenced by:: ¶ ‣ §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E2
Encodings:: TeX, pMML, png

22.7.3 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\frac{\mathop{\mathrm{cn}\/}\nolimits\left(z/(1+k_{1}),k_{1}\right)\mathop{\mathrm{dn}\/}\nolimits\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z/(1+k_{1}),k_{1}\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.12.3
Referenced by:: ¶ ‣ §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E3
Encodings:: TeX, pMML, png

22.7.4 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=\frac{{\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z/(1+k_{1}),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z/(1+k_{1}),k_{1}\right)}.$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.12.4
Referenced by:: ¶ ‣ §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E4
Encodings:: TeX, pMML, png

§22.7(ii) Ascending Landen Transformation

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

With

22.7.5

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

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

Symbols:: $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.14.1
Permalink:: http://dlmf.nist.gov/22.7.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

22.7.6 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\frac{(1+k^{{\prime}}_{2})\mathop{\mathrm{sn}\/}\nolimits\left(z/(1+k^{{\prime}}_{2}),k_{2}\right)\mathop{\mathrm{cn}\/}\nolimits\left(z/(1+k^{{\prime}}_{2}),k_{2}\right)}{\mathop{\mathrm{dn}\/}\nolimits\left(z/(1+k^{{\prime}}_{2}),k_{2}\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.14.2
Permalink:: http://dlmf.nist.gov/22.7.E6
Encodings:: TeX, pMML, png

22.7.7 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\frac{(1+k^{{\prime}}_{2})({\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z/(1+k^{{\prime}}_{2}),k_{2}\right)-k^{{\prime}}_{2})}{k_{2}^{2}\mathop{\mathrm{dn}\/}\nolimits\left(z/(1+k^{{\prime}}_{2}),k_{2}\right)},$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.14.3
Permalink:: http://dlmf.nist.gov/22.7.E7
Encodings:: TeX, pMML, png

22.7.8 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=\frac{(1-k^{{\prime}}_{2})({\mathop{\mathrm{dn}\/}\nolimits^{{2}}}\left(z/(1+k^{{\prime}}_{2}),k_{2}\right)+k^{{\prime}}_{2})}{k_{2}^{2}\mathop{\mathrm{dn}\/}\nolimits\left(z/(1+k^{{\prime}}_{2}),k_{2}\right)}.$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.14.4
Permalink:: http://dlmf.nist.gov/22.7.E8
Encodings:: TeX, pMML, png

§22.7(iii) Generalized Landen Transformations

Keywords:: Jacobian elliptic functions, Landen transformations
Permalink:: http://dlmf.nist.gov/22.7.iii

See Khare and Sukhatme (2004).

§22.7 Landen Transformations

Contents

§22.7(i) Descending Landen Transformation

§22.7(ii) Ascending Landen Transformation

§22.7(iii) Generalized Landen Transformations