Recollection of the role of Liouville in the history of the elliptic functions

Abstract

This paper collects the main conclusions of a research on the contributions of J. Liouville to the contemporary theory of the elliptic functions. It covers most of the results of a collaboration between the SUMMA group from the Universidad de Medellín Basic Sciences Department and the Mat group from the Universidad del Tolima Department of Mathematics and Statistics. The project has been partially funded by the Universidad de Medellín Research Vice-Principal’s Office and the Universidad del Tolima School of Sciences. It begins with a description of Liouville’s historic background after the emergence of the elliptic function modern concept in the work of
Abel and Jacobi. Subsequently, certain details of the Leçons chaired by the famous French mathematician in 1847 are discussed. Such details cover the so-called Liouville-Borchardt theorem, the fundamental propositions about the number of zeros of the meromorphic doubly periodic functions, and the results of the relation between the zeros and the poles. The end of the article outlines important conclusions regarding the Liouville’s legacy to the current theory of the elliptic functions.