Chapter 4 Complex Functions and Elliptic Integrals

Abstract

In this chapter we consider how elliptic function theory and complex variable theory were finally drawn together in the 1830s and 1840s. As the recognition of the importance of the work of Abel and Jacobi grew, mathematicians came to feel that it was unsatisfactory to base the theory of elliptic functions on the inversion of many-valued integrals. One alternative would have been to adopt and develop Cauchy’s theory of complex integrals. By and large this was not done, and it is interesting to examine why. The study of elliptic integrals was felt by many to be fraught with ambiguity because of the square root in the integrand. Moreover, Cauchy’s system of definitions, based on his newly defined concepts of limit, continuity, differentiability, and integrability, was incompatible with talk of many-valued functions—Cauchy did not define continuity for a many-valued function, and indeed a many-valued function cannot be continuous according to Cauchy’s use of the term. Although a doubly periodic function is a meromorphic function defined on the whole of the complex plane, an elliptic integral makes better sense on something like a Riemann surface (a torus in this case). Thus the many-valued nature of an elliptic integral posed a challenge to mathematicians throughout the 1830s and 1840s. So the perceived problem with the foundations did not meet with a ready answer in the newly emerging theory of complex functions. Matters were to be worse with hyperelliptic integrals, because the corresponding inverse functions could not be treated as multiply-periodic functions in the plane.