A characterization of Herglotz-Nevanlinna functions in two variables via integral representations

TL;DR

This paper provides a comprehensive integral representation for Herglotz-Nevanlinna functions in two variables, characterizing them completely through specific real, non-negative, and measure-based parameters.

Contribution

It introduces a new integral representation that fully characterizes two-variable Herglotz-Nevanlinna functions using measure theory and parameter conditions.

Findings

Complete characterization of two-variable Herglotz-Nevanlinna functions

Derivation of integral representation involving measures

Discussion of properties of representing measures

Abstract

We derive an integral representation for Herglotz-Nevanlinna functions in two variables which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the representing measures are discussed.