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22 Jacobian Elliptic FunctionsProperties

§22.11 Fourier and Hyperbolic Series

Throughout this section q and ζ are defined as in §22.2.

If qexp(2|ζ|)<1, then

22.11.1 sn(z,k) =2πKkn=0qn+12sin((2n+1)ζ)1-q2n+1,
22.11.2 cn(z,k) =2πKkn=0qn+12cos((2n+1)ζ)1+q2n+1,
22.11.3 dn(z,k) =π2K+2πKn=1qncos(2nζ)1+q2n.
22.11.4 cd(z,k) =2πKkn=0(-1)nqn+12cos((2n+1)ζ)1-q2n+1,
22.11.5 sd(z,k) =2πKkkn=0(-1)nqn+12sin((2n+1)ζ)1+q2n+1,
22.11.6 nd(z,k) =π2Kk+2πKkn=1(-1)nqncos(2nζ)1+q2n.

Next, if qexp(|ζ|)<1, then

22.11.7 ns(z,k)-π2Kcscζ =2πKn=0q2n+1sin((2n+1)ζ)1-q2n+1,
22.11.8 ds(z,k)-π2Kcscζ =-2πKn=0q2n+1sin((2n+1)ζ)1+q2n+1,
22.11.9 cs(z,k)-π2Kcotζ =-2πKn=1q2nsin(2nζ)1+q2n,
22.11.10 dc(z,k)-π2Ksecζ=2πKn=0(-1)nq2n+1cos((2n+1)ζ)1-q2n+1,
22.11.11 nc(z,k)-π2Kksecζ=-2πKkn=0(-1)nq2n+1cos((2n+1)ζ)1+q2n+1,
22.11.12 sc(z,k)-π2Kktanζ=2πKkn=1(-1)nq2nsin(2nζ)1+q2n.

In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions.

Next, with E=E(k) denoting the complete elliptic integral of the second kind (§19.2(ii)) and qexp(2|ζ|)<1,

22.11.13 sn2(z,k)=1k2(1-EK)-2π2k2K2n=1nqn1-q2ncos(2nζ).

Similar expansions for cn2(z,k) and dn2(z,k) follow immediately from (22.6.1).

For further Fourier series see Oberhettinger (1973, pp. 23–27).

A related hyperbolic series is

22.11.14 k2sn2(z,k)=EK-(π2K)2n=-(sech2(π2K(z-2nK))),

where E=E(k) is defined by §19.2.9. Again, similar expansions for cn2(z,k) and dn2(z,k) may be derived via (22.6.1). See Dunne and Rao (2000).