Jacobian elliptic functions with real moduli in the intervals and , or with purely imaginary moduli are related to functions with moduli in the interval by the following formulas.
for all twelve functions.
When is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of . For illustrations see Figures 22.3.25–22.3.29. In consequence, the formulas in this chapter remain valid when is complex. In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of , irrespective of which values of and are chosen—as long as they are used consistently. For proofs of these results and further information see Walker (2003).