Non-elementary functions

Abstract

The theory of the preceding chapters permits the construction and investigation of new transcendental functions. The Γ-function, interpolating the factorials, is perhaps the most important nonelementary function (V.1). Riemann's ζ-function (V.2,3) and its generalizations play an eminent role in number theory and algebraic geometry. It is the main tool in most proofs of the prime number theorem (V.2). Elliptic functions, i.e. functions with two independent periods, and their connection with plane cubic curves is a classical theme (V.4,5) with applications in many areas, e.g. mathematical physics and cryptography.