§22.17 Moduli Outside the Interval [0,1]

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

First

for all twelve functions.

Secondly,

Thirdly, with

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.

When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^2$. For illustrations see Figures 22.3.25–22.3.29. 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 $\sqrt{k}$ and $k^{\prime }=\sqrt{1-k^2}$ are chosen—as long as they are used consistently. For proofs of these results and further information see Walker (2003).