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22 Jacobian Elliptic FunctionsProperties

§22.17 Moduli Outside the Interval [0,1]

Contents

§22.17(i) Real or Purely Imaginary Moduli

Jacobian elliptic functions with real moduli in the intervals (-,0) and (1,), or with purely imaginary moduli are related to functions with moduli in the interval [0,1] by the following formulas.

First

22.17.1 pq(z,k)=pq(z,-k),

for all twelve functions.

Secondly,

22.17.2 sn(z,1/k) =ksn(z/k,k),
22.17.3 cn(z,1/k) =dn(z/k,k),
22.17.4 dn(z,1/k) =cn(z/k,k).

Thirdly, with

22.17.5 k1 =k1+k2,
k1k1 =k1+k2,
22.17.6 sn(z,ik) =k1sd(z/k1,k1),
22.17.7 cn(z,ik) =cd(z/k1,k1),
22.17.8 dn(z,ik) =nd(z/k1,k1).

In terms of the coefficients of the power series of §22.10(i), the above equations are polynomial identities in k. In (22.17.5) either value of the square root can be chosen.

§22.17(ii) Complex Moduli

When z is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of k2. For illustrations see Figures 22.3.2522.3.29. In consequence, the formulas in this chapter remain valid when k is complex. In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of k, irrespective of which values of k and k=1-k2 are chosen—as long as they are used consistently. For proofs of these results and further information see Walker (2003).