22.6 Elementary Identities22.8 Addition Theorems

§22.7 Landen Transformations

Contents

§22.7(i) Descending Landen Transformation

With

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

§22.7(ii) Ascending Landen Transformation

§22.7(iii) Generalized Landen Transformations

See Khare and Sukhatme (2004).