Introduction: the extension problem

In this chapter we define the Jacobian elliptic functions and establish their basic properties; for the convenience of the reader a summary of those properties is included at the end of the chapter.

In the last section of Chapter 1 we sought to make clear the possibility that the Jacobian elliptic functions (arising out of our study of the simple pendulum and as defined in (1.22) of that chapter) are much more than routine generalizations of the circular functions. But to explore their suspected richness it is necessary to move off the real line into the complex plane.

Our treatment is based on Abel's original insight (1881), and is in three steps: (i) extension of the definitions, (1.28), to the imaginary axis; (ii) derivation of the addition formulae for the functions sn, cn and dn; (iii) formal extension to the complex plane by means of the addition formulae. The discussion in Section 2.6 then shows that the extended functions are doubly periodic.

What appears to be unusual (perhaps new) in our treatment is: step (iv), verification that the functions so extended are analytic except for poles (that is, meromorphic) in the finite ℂ plane. (Of course, there are other ways of obtaining the functions sn, cn and dn as meromorphic functions of a complex variable, but none of them is easy.