Degenerate Abelian Integrals

Abstract

Kovalevskaya’s work on Abelian integrals which reduce to elliptic integrals (1884) is the hardest of her papers to explain to a general audience since it assumes a detailed knowledge of the nineteenth-century work in this area. The subject of Abelian integrals in the nineteenth century was a vast corpus of results, many of which are not generally taught nowadays, even to specialists in algebraic functions. This chapter therefore requires more preliminary explanation than the other chapters in this book, partly in order to introduce the reader properly to the mathematical concepts involved and partly in order to establish the historical context for Kovalevskaya’s work. Appendix 2 contains a little analytic function theory for readers who have not had such a course. It is hoped that this appendix may clarify any mathematical obscurities in the exposition. Naturally this chapter is not a textbook on Abelian integrals. The reader who does not know the subject already should nevertheless be able to appreciate in general terms the significance of the problem Kovalevskaya worked on. A really detailed exposition of the subject, even with mathematical details omitted, would have to be much longer than the present chapter. As a matter of fact, several such expositions were written around the beginning of the twentieth century, for instance Brill and Noether (1894) and Krazer and Wirtinger (1921). The latter occupies nearly 300 pages, proofs omitted.