Value Distribution Theory and Related Topics - Barsegian, G.; Laine, I.; Yang, C. C. (Springer US, 2004)
The Nevanlinna theory of value distribution of meromorphic functions, one of the milestones of complex analysis during the last century, was created to extend the classical results concerning the distribution of of entire functions to the more general setting of meromorphic functions. Later on, a similar reasoning has been applied to algebroid functions, subharmonic functions and meromorphic functions on Riemann surfaces as well as to analytic functions of several complex variables, holomorphic and meromorphic mappings and to the theory of minimal surfaces. Moreover, several applications of the theory have been exploited, including complex differential and functional equations, complex dynamics and Diophantine equations. The main emphasis of this collection is to direct attention to a number of recently developed novel ideas and generalizations that relate to the development of value distribution theory and its applications. In particular, we mean a recent theory that replaces the conventional consideration of counting within a disc by an analysis of their geometric locations. Another such example is presented by the generalizations of the second main theorem to higher dimensional cases by using the jet theory. Moreover, similar ideas apparently may be applied to several related areas as well, such as to partial differential equations and to differential geometry. Indeed, most of these applications go back to the problem of analyzing zeros of certain complex or real functions, meaning in fact to investigate level sets or level surfaces