Cauchy elliptic and Abelian integrals

The mathematical problem with which researchers were confronted can be simply stated. Cauchy' s whole system of definitions, based on his newly-defined concepts of limit, continuity, differentiability, and integrability, was incompatible with talk of many-valued functions. A many-valued function (to use the 19th century term) is nowhere continuous in Cauchy's sense. But even a naive treatment of elliptic integrals was felt by many to be fraught with ambiguity because of the square root in the integrand. While a doubly periodic function is a meromorphic function defined on the whole of the complex plane, an elliptic integral only makes sense on something like a Riemann surface (which is a torus in this case). Thus the many-valued nature of an elliptic integral posed a challenge to mathematicians throughout the 1830s and 1840s.

Rev. Hist. Sci., 1992, XLV/1