On applications of Herglotz-Nevanlinna functions in material sciences, I: classical theory and applications of sum rules

TL;DR

This review introduces the theory of Herglotz-Nevanlinna functions and explores their applications in material sciences, particularly in electromagnetics, focusing on sum rules and foundational properties.

Contribution

It provides a comprehensive overview of scalar, matrix, and operator-valued Herglotz-Nevanlinna functions and their relevance to material science applications, especially electromagnetics.

Findings

Detailed theorems relevant to applications

Overview of matrix and operator-valued functions

Applications related to sum rules in electromagnetics

Abstract

This is the first part of the review article which focuses on theory and applications of Herglotz-Nevanlinna functions in material sciences. It starts with the definition of scalar valued Herglotz-Nevanlinna functions and explains in detail the theorems that are pertinent to applications, followed by a short overview of the matrix-valued and operator-valued versions of these functions and the properties that carry over from scalar cases. The theory is complemented by some applications from electromagnetics that are related to the sum rules. More applications of Herglotz Nevanlinnna functions in material sciences can be found in Part II.