The Elliptic Modular Function and a Class of Analytic Functions First Considered by Hurwitz

Abstract

Introduction. In spite of its wide applicability in various branches of the theory of functions, the elliptic modular function is often used with a certain hesitation. This is mainly due to the fact that its application presupposes familiarity with a comparatively intricate formalism, in particular when the determination of numerical constants is involved. In fact, the endeavor to avoid the elliptic modular function has given rise to an extensive mathematical literature aiming at proving certain theorems in an way, the word being used here as a synonym for without making use of the elliptic modular function. As an impressive example, Picard's theorem on integral functions might be quoted. The difficulties which beset the numerical treatment of the elliptic modular function go essentially back to the fact that, on the one hand, the formalism of this function can only be developed with the help of the Jacobian elliptic functions while, on the other hand, what is needed in the applications are the conformal mapping properties of the modular function, and the connection between these two different aspects of the modular function has to be established through the medium of the theory of Schwarz' differential parameter or by a very detailed study of the periodic properties of the Jacobian elliptic functions. The object of the first part of this paper is to show how those properties of the elliptic modular function which are required for the applications may be derived in a simple way by the exclusive use of elementary principles of the theory of conformal representation. It will be shown that once the modular surface' is defined, the functional equation