# §22.7 Landen Transformations

## §22.7(i) Descending Landen Transformation

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

With

 22.7.5 $\displaystyle k_{2}$ $\displaystyle=\frac{2\sqrt{k}}{1+k},$ $\displaystyle k_{2}^{\prime}$ $\displaystyle=\frac{1-k}{1+k},$ ⓘ 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 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},$
 22.7.7 $\operatorname{cn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)-k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)},$
 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

See Khare and Sukhatme (2004).