Orientation at singularities of harmonic functions
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Orientation at singularities of harmonic functions Juan Arango1 · Hugo Arbeláez1
· Jheison Rivera1
Received: 25 April 2020 / Accepted: 18 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We find a simple expression in complex terms for homogeneous harmonic polynomials, which we use to express the Laurent series of a harmonic function around an isolated singularity. Also, we show a residue theorem and study the orientation at isolated singularities through the use of complex dilatation, focusing on those points where orientation is not preserved nor reversed, making essential the concept of exceptional set and extending it to isolated singularities. Keywords Harmonic mappings · Homogeneous polynomials · Isolated singularities · Laurent series · Residue theorem · Exceptional set Mathematics Subject Classification 30C10 · 30B10 · 31A05
1 Introduction The argument principle is an important and useful result for meromorphic functions on domains in the plane. Duren, Hengartner and Laugesen have obtained an argument principle for harmonic mappings on Jordan domains [5]. Years later, Suffridge and Thompson proved an argument principle for harmonic mappings that have isolated singularities [9]; also included in the paper is a definition of “pole” at an isolated singular point of a harmonic mapping and a “partitioning theorem” for the image space that yields components whose values are assumed the same number of times in the
Communicated by Adrian Constantin.
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Hugo Arbeláez [email protected] Juan Arango [email protected] Jheison Rivera [email protected]
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Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellín, Medellín, Colombia
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J. Arango et al.
appropriate region. Although harmonic mappings (i.e., univalent harmonic functions) have been extensively studied in recent decades, see for example [1,3,4] and [6], the topic of harmonic functions with isolated singularities has not been considered by many authors. In this paper we will approach this topic, emphasizing in the orientation of a harmonic function around an isolated singularity. In a simply connected domain Ω ⊆ C, a complex-valued harmonic function f has the representation f = h + g, where h and g are analytic in Ω; this representation is unique up to an additive constant. If f is real-valued, the representation reduces to f = h + h = Re{2h}, where 2h is the analytic completion of f , unique up to an additive imaginary constant. Since we are interested in studying harmonic functions that are not necessarily univalent, the Jacobian of these functions is a useful tool. For analytic functions f , it is a classical result that J f (z) = 0 if and only if f is locally univalent at z. Hans Lewy showed in 1936 [8] that this remains true for harmonic mappings. In view of Lewy’s theorem, harmonic mappings are either sense-preserving with J f (z) > 0, or sense-reversing with J f (z) < 0 throughout the domain Ω where f is univalent. If f is sense-preserving, then f is sense-reversing. The Jacobian of a func
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