During the decade and a half that has elapsed since the intro
duction of principal functions (Sario 8 J), they have become impor
tant tools in an increasing number of branches of modern mathe
matics. The purpose of the present research monograph is to
systematically develop the theory of these functions and their ap
plications on Riemann surfaces and Riemannian spaces. Apart from
brief background information (see below), nothing contained in this
monograph has previously appeared in any other book. The basic idea
of principal functions is simple: Given a Riemann surface or a
Riemannian space R, a neighborhood A of its ideal boundary, and a
harmonic function s on A, the principal function problem consists
in constructing a harmonic function p on all of R which imitates
the behavior of s in A. Here A need not be connected, but may
include neighborhoods of isolated points deleted from R. Thus we
are dealing with the general problem of constructing harmonic
functions with given singularities and a prescribed behavior near
the ideal boundary. The function p is called the principal function
corresponding to the given A, s, and the mode of imitation of s by
p. The significance of principal functions is in their
versatility."
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