Doubly Periodic Functions

Abstract

In the mathematical community of today, Liouville’s name is primarily attached to the theory of complex functions. The reason is that among the many mathematical theorems named after him, the most celebrated is the one stating that a bounded holomorphic function boldmath C → boldmath C is necessarily constant. It is less well known that Liouville conceived of this theorem as the basis of an elegant approach to elliptic functions. The basic element of this approach is to deduce the properties of elliptic functions from their double periodicity, rather than from their representation as inverses of elliptic integrals. After Liouville this has become the standard approach in elementary textbooks to the extent that an elliptic function is nowadays usually defined as a doubly periodic meromorphic function. In this chapter, however, I shall follow Liouville and call such functions doubly periodic, whereas the term “elliptic function” will be used in the Jacobian sense to denote the inverse of an elliptic integral. Considering the great impact of Liouville’s ideas on the theory of doubly periodic functions, a biography of Liouville could hardly be considered complete if this subject was left out. Therefore, I have included this chapter despite the fact that Jeanne Peiffer has recently published an excellent paper on this subject [Peiffer 1983]. Another justification for this chapter is that I shall emphasize other aspects of the story than did Jeanne Peiffer, such as Liouville’s early paper on the division of elliptic functions, the chronological development of his ideas on doubly periodic functions, and the essential elements of his final approach. In particular, I shall reconstruct Liouville’s first proof of Liouville’s theorem, a proof which she only implicitly mentioned. On the other hand, I shall gloss over Liouville’s ideas on the foundation of complex analysis and their relation to the ideas of his contemporaries, an intricate question discussed in detail by Peiffer.