Appears in collection : Herglotz-Nevanlinna Functions and their Applications to Dispersive Systems and Composite Materials / Fonctions de Herglotz-Nevanlinna et leurs applications aux systèmes dispersifs et aux matériaux composites

In this survey talk we are first going to recall several equivalent (but partly still quite different) ways to characterize the class of Herglotz-Nevanlinna functions, and discuss the corresponding representations and tools. Based on this point of view there are different ways to extend the class of Herglotz-Nevanlinna functions by generalizing the defining properties. In this talk we are going to discuss few of them, with special focus on wether classical tools are still available or where they can fail.