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22 Jacobian Elliptic FunctionsProperties

§22.6 Elementary Identities

Contents

§22.6(i) Sums of Squares

22.6.1 sn2(z,k)+cn2(z,k)=k2sn2(z,k)+dn2(z,k)=1,
22.6.2 1+cs2(z,k)=k2+ds2(z,k)=ns2(z,k),
22.6.3 k2sc2(z,k)+1=dc2(z,k)=k2nc2(z,k)+k2,
22.6.4 -k2k2sd2(z,k)=k2(cd2(z,k)-1)=k2(1-nd2(z,k)).

§22.6(ii) Double Argument

22.6.5 sn(2z,k)=2sn(z,k)cn(z,k)dn(z,k)1-k2sn4(z,k),
22.6.6 cn(2z,k)=cn2(z,k)-sn2(z,k)dn2(z,k)1-k2sn4(z,k)=cn4(z,k)-k2sn4(z,k)1-k2sn4(z,k),
22.6.7 dn(2z,k)=dn2(z,k)-k2sn2(z,k)cn2(z,k)1-k2sn4(z,k)=dn4(z,k)+k2k2sn4(z,k)1-k2sn4(z,k).
22.6.8 cd(2z,k) =cd2(z,k)-k2sd2(z,k)nd2(z,k)1+k2k2sd4(z,k),
22.6.9 sd(2z,k) =2sd(z,k)cd(z,k)nd(z,k)1+k2k2sd4(z,k),
22.6.10 nd(2z,k) =nd2(z,k)+k2sd2(z,k)cd2(z,k)1+k2k2sd4(z,k),
22.6.11 dc(2z,k) =dc2(z,k)+k2sc2(z,k)nc2(z,k)1-k2sc4(z,k),
22.6.12 nc(2z,k) =nc2(z,k)+sc2(z,k)dc2(z,k)1-k2sc4(z,k),
22.6.13 sc(2z,k) =2sc(z,k)dc(z,k)nc(z,k)1-k2sc4(z,k),
22.6.14 ns(2z,k) =ns4(z,k)-k22cs(z,k)ds(z,k)ns(z,k),
22.6.15 ds(2z,k) =k2k2+ds4(z,k)2cs(z,k)ds(z,k)ns(z,k),
22.6.16 cs(2z,k) =cs4(z,k)-k22cs(z,k)ds(z,k)ns(z,k).

See also Carlson (2004).

22.6.17 1-cn(2z,k)1+cn(2z,k) =sn2(z,k)dn2(z,k)cn2(z,k),
22.6.18 1-dn(2z,k)1+dn(2z,k) =k2sn2(z,k)cn2(z,k)dn2(z,k).

§22.6(iii) Half Argument

22.6.19 sn2(12z,k) =1-cn(z,k)1+dn(z,k)
=1-dn(z,k)k2(1+cn(z,k))
=dn(z,k)-k2cn(z,k)-k2k2(dn(z,k)-cn(z,k)),
22.6.20 cn2(12z,k) =-k2+dn(z,k)+k2cn(z,k)k2(1+cn(z,k))
=k2(1-dn(z,k))k2(dn(z,k)-cn(z,k))
=k2(1+cn(z,k))k2+dn(z,k)-k2cn(z,k),
22.6.21 dn2(12z,k) =k2cn(z,k)+dn(z,k)+k21+dn(z,k)
=k2(1-cn(z,k))dn(z,k)-cn(z,k)
=k2(1+dn(z,k))k2+dn(z,k)-k2cn(z,k).

If {p,q,r} is any permutation of {c,d,n}, then

22.6.22 pq2(12z,k)=ps(z,k)+rs(z,k)qs(z,k)+rs(z,k)=pq(z,k)+rq(z,k)1+rq(z,k)=pr(z,k)+1qr(z,k)+1.

For (22.6.22) and similar results, see Carlson (2004).

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.
sn(iz,k)= isc(z,k) dc(iz,k)= dn(z,k)
cn(iz,k)= nc(z,k) nc(iz,k)= cn(z,k)
dn(iz,k)= dc(z,k) sc(iz,k)= isn(z,k)
cd(iz,k)= nd(z,k) ns(iz,k)= -ics(z,k)
sd(iz,k)= isd(z,k) ds(iz,k)= -ids(z,k)
nd(iz,k)= cd(z,k) cs(iz,k)= -ins(z,k)

§22.6(v) Change of Modulus

See §22.17.