  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.
