Digital Library of Mathematical Functions
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22 Jacobian Elliptic FunctionsProperties

§22.7 Landen Transformations

Contents

§22.7(i) Descending Landen Transformation

With

22.7.1 k1=1-k1+k,
22.7.2 sn(z,k)=(1+k1)sn(z/(1+k1),k1)1+k1sn2(z/(1+k1),k1),
22.7.3 cn(z,k)=cn(z/(1+k1),k1)dn(z/(1+k1),k1)1+k1sn2(z/(1+k1),k1),
22.7.4 dn(z,k)=dn2(z/(1+k1),k1)-(1-k1)1+k1-dn2(z/(1+k1),k1).

§22.7(ii) Ascending Landen Transformation

With

22.7.5 k2 =2k1+k,
k2 =1-k1+k,
22.7.6 sn(z,k)=(1+k2)sn(z/(1+k2),k2)cn(z/(1+k2),k2)dn(z/(1+k2),k2),
22.7.7 cn(z,k)=(1+k2)(dn2(z/(1+k2),k2)-k2)k22dn(z/(1+k2),k2),
22.7.8 dn(z,k)=(1-k2)(dn2(z/(1+k2),k2)+k2)k22dn(z/(1+k2),k2).

§22.7(iii) Generalized Landen Transformations

See Khare and Sukhatme (2004).