A Method of Deriving the Infinite Double Products in the Theory of Elliptic Functions from the Multiplication Theorems

Abstract

In the memoir En metod att komma i analytisk besittning af de elliptiska funktionernat we have shown that the method given by Abel in his Recherches sur les fonctions elliptiques,+ which is historically the earliest method of deriving the elliptic functions, is also mathematically sound and leads to the desired result with perfect rigor. The method of Abel has the advantage over all other methods that it does not presuppose any theorems from the general theory of functions. It is consequently the most elementary of all mathematical processes which lead to a real command of the analytical properties of the elliptic functions. A few serious objections can be raised against Abel's own presentation, however. The most important of these is that the passage from the multiplication theorems to the infinite double series and double products is not sufficiently justified; in fact, it is based upon an argument which it is scarcely possible to make mathematically rigorous. This has been pointed out by Broch in his paper Om de elliptiske Funktioners Roekkeudvikling.? In this paper, as well as in his learned and suggestive book Traite Elementaire des Fonctions Elliptiques, ff there is developed a new derivation of this limiting passage which is perfectly distinct from that of Abel.