Valence of Complex-valued Planar Harmonic Functions
نویسنده
چکیده
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 ⊆ R\f(f(S) ∪ C(f)) is a n0-fold covering of some component Ω0 ⊆ C\(f(S) ∪ C(f)). If Ω0 is simply connected, then f is univalent on R0. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for C functions on open sets in R are first stated in that form and then applied to the case of planar harmonic functions. 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).
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