On multiple uses of some classical equations

Abstract

In Complex and Real Analysis many classes of important functions can be defined as solutions of functional equations which satisfy certain regularity conditions (e.g. analyticity or holomorphy in the complex case, convexity in the real case). A striking example was the restructuring of the theory of the gamma function by E. Artin which made the theory in the real case crystal clear and esthetically pleasing and served as a pattern for developing similar theories for other special functions. In complex analysis we may consider the theory of automorphic and modular functions as a chapter which is governed by systems of functional equations, and this is even more evident for the zeta functions (in number theory and algebraic geometry). In the investigation of these functions their functional equations play a decisive role since all interesting analytic properties of these functions are derived by studying the equations. Just recently, the dilogarithm became interesting in algebraic topology and in topology of 3-dimensional manifolds. Many of the functional equations satisfied by the dilogarithm are now of use for this purpose, after having been forgotten for years.