# §22.7 Landen Transformations

## §22.7(i) Descending Landen Transformation

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 See also: Annotations for 22.7(i)
 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: $\mathop{\mathrm{sn}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: 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 See also: Annotations for 22.7(i)
 22.7.3 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\frac{\mathop{\mathrm{cn}\/}% \nolimits\left(z/(1+k_{1}),k_{1}\right)\mathop{\mathrm{dn}\/}\nolimits\left(z/% (1+k_{1}),k_{1}\right)}{1+k_{1}{\mathop{\mathrm{sn}\/}\nolimits^{2}}\left(z/(1% +k_{1}),k_{1}\right)},$
 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: $\mathop{\mathrm{dn}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: 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 See also: Annotations for 22.7(i)

## §22.7(ii) Ascending Landen Transformation

With

 22.7.5 $\displaystyle k_{2}$ $\displaystyle=\frac{2\sqrt{k}}{1+k},$ $\displaystyle k^{\prime}_{2}$ $\displaystyle=\frac{1-k}{1+k},$ 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 See also: Annotations for 22.7(ii)
 22.7.6 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\frac{(1+k^{\prime}_{2})% \mathop{\mathrm{sn}\/}\nolimits\left(z/(1+k^{\prime}_{2}),k_{2}\right)\mathop{% \mathrm{cn}\/}\nolimits\left(z/(1+k^{\prime}_{2}),k_{2}\right)}{\mathop{% \mathrm{dn}\/}\nolimits\left(z/(1+k^{\prime}_{2}),k_{2}\right)},$
 22.7.7 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\frac{(1+k^{\prime}_{2})({% \mathop{\mathrm{dn}\/}\nolimits^{2}}\left(z/(1+k^{\prime}_{2}),k_{2}\right)-k^% {\prime}_{2})}{k_{2}^{2}\mathop{\mathrm{dn}\/}\nolimits\left(z/(1+k^{\prime}_{% 2}),k_{2}\right)},$
 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: $\mathop{\mathrm{dn}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$: 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 See also: Annotations for 22.7(ii)

## §22.7(iii) Generalized Landen Transformations

See Khare and Sukhatme (2004).