Valence of harmonic functions

This dissertation concerns questions related to the valence of complex-valued planar harmonic functions. The valence of a function f at a point w is the number of distinct, finite solutions to f(z) = w. Let f be a complex-valued harmonic function in an open set R ⊆ C. Let S denote the critical set of f and C(f) the global cluster set of f. We show that f(S) ∪ C(f) partitions the complex plane into regions of constant valence. We give some conditions such that f( S) ∪ C(f) has empty interior. We also show that a component R0 ⊆ Rf-1(f( S) ∪ C(f)) is a n 0-fold covering of some component O0 ⊆ C(f(S) ∪ C( f)). If O0 is simply connected, then f is univalent on R0. We explore conditions for combining adjacent components to form a larger region of univalence. If f is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of C(f(S) ∪ C(f)) sharing a common boundary arc in f(S)C(f). We explore the behavior of f on analytic arcs which meet the critical set at a point. We also extend a result of Wilmshurst concerning conditions for a harmonic polynomial to have finite valence.