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22 Jacobian Elliptic FunctionsProperties

§22.6 Elementary Identities

Contents
  1. §22.6(i) Sums of Squares
  2. §22.6(ii) Double Argument
  3. §22.6(iii) Half Argument
  4. §22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
  5. §22.6(v) Change of Modulus

§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(1nd2(z,k)).

§22.6(ii) Double Argument

22.6.5 sn(2z,k)=2sn(z,k)cn(z,k)dn(z,k)1k2sn4(z,k),
22.6.6 cn(2z,k)=cn2(z,k)sn2(z,k)dn2(z,k)1k2sn4(z,k)=cn4(z,k)k2sn4(z,k)1k2sn4(z,k),
22.6.7 dn(2z,k)=dn2(z,k)k2sn2(z,k)cn2(z,k)1k2sn4(z,k)=dn4(z,k)+k2k2sn4(z,k)1k2sn4(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)1k2sc4(z,k),
22.6.12 nc(2z,k) =nc2(z,k)+sc2(z,k)dc2(z,k)1k2sc4(z,k),
22.6.13 sc(2z,k) =2sc(z,k)dc(z,k)nc(z,k)1k2sc4(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 1cn(2z,k)1+cn(2z,k) =sn2(z,k)dn2(z,k)cn2(z,k),
22.6.18 1dn(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) =1cn(z,k)1+dn(z,k)=1dn(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(1dn(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(1cn(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.