Hidden Harmony—Geometric Fantasies

On 28 June 1830 the Académie des sciences in Paris, the leading scientific institution of the day, announced that its Grand Prize in mathematics of 3,000 francs devoted to work which “presents the most important application of mathematical theories …or which contains a very remarkable analytical discovery” would be divided equally between Carl Gustav Jacob Jacobi in Königsberg and the family of the late Niels Henrik Abel of Christiania.

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.

Mathematical creativity is often said to be peculiar to young people. Counterexamples to this widespread opinion are indeed quite rare in the history of mathematics, but Weierstrass is one of them, and possibly the most remarkable. Even when he was 39, the age at which Riemann died, Weierstrass was still an unknown school teacher. Admittedly, by that time he had already written some five papers. Most of them, however, had remained in manuscript or had been printed in virtually unknown school programs.

By 1880 the main features of the present-day complex function theory had been created, although not assembled into their modern order. Cauchy and Riemann were dead, but Weierstrass was in firm control of his own version of the theory and lecturing on it as part of his 2-year cycle of lectures in Berlin. As we saw in the previous chapter, complex function theory in its various forms had found in widening number of successful applications to other domains of both pure and applied mathematics. In this chapter we look at how the ideas of the founders led to important developments in the theory of complex functions itself and to the discovery of some of the central features of complex function theory as a rich research topic in its own right.

Throughout the nineteenth century and well into the twentieth the study of complex functions of several variables posed a challenge to the experts in the function theory of a single variable. As we have seen in Chap. 6, the prospect of creating a theory of Abelian functions was one that Weierstrass continually had in mind; it was the ultimate goal of all his work. And yet a marked distinction between the theories of one and several variables persists to the present day. Almost all universities offer a mainstream course in single variable complex function theory; few, if any, present the theory of several variables as other than a specialist option. We shall see that this distinction is in the nature of the functions studied. Because this dichotomy survives in the modern syllabus, we have divided this chapter into three sections. The first is a survey of the claim that the theories of one and several variables diverge markedly. We give an indication of what was discovered about the complex function theory of several variables, but generally slight the proofs so that the section can be read by a broad audience. The second section looks at the history of the principal results about Abelian functions and theta functions which, for a long time, were the only examples known of complex functions of several variables. In the final section we reconsider the general theory, look at some of the techniques used, and seek sharpen the discussion of the opening section. The latter sections naturally place greater demands on the reader.

It is a significant event in the life of a mathematical topic when it enters the student degree syllabus, and another when it does so in a generally agreed way in many different universities. Today a remarkable degree of consensus exists about what constitutes elementary complex function theory. It is the purpose of this chapter to investigate how this consensus came about and in what way it embodies a vision of complex function theory central to the modern mathematical syllabus.