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.
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References
Aleman, A., Constantin, A.: Harmonic maps and ideal fluid flows. Arch. Ration. Mech. Anal. 204, 479–513 (2012)
Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer-Verlag, New York (2001)
Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A.I. 9, 3–25 (1984)
Constantin, A., Martin, M.J.: A harmonic maps approach to fluid flows. Math. Ann. 369, 1–16 (2017)
Duren, P., Hengartner, W., Laugesen, R.: The argument principle for harmonic functions. Am. Math. Monthly. 103, 411–415 (1996)
Duren, P.: Harmonic Mappings in the Plane. Cambridge University Press, Cambridge (2004)
Hengartner, W., Schober, G.: Univalent harmonic functions. Trans. Am. Math. Soc. 299, 1–31 (1987)
Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)
Suffridge, T.J., Thompson, J.W.: Local behavior of harmonic mappings. Complex Var. Theory Appl. 41, 63–80 (2000)
Acknowledgements
The authors want to thank the Universidad Nacional de Colombia, for the support for this Project. The first and second authors were supported by the Universidad Nacional de Colombia, Hermes Project 49148. The third author wishes to thank the support by Colciencias for the scholarship 6172.
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Communicated by Adrian Constantin.
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Arango, J., Arbeláez, H. & Rivera, J. Orientation at singularities of harmonic functions. Monatsh Math 193, 737–762 (2020). https://doi.org/10.1007/s00605-020-01441-4
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DOI: https://doi.org/10.1007/s00605-020-01441-4
Keywords
- Harmonic mappings
- Homogeneous polynomials
- Isolated singularities
- Laurent series
- Residue theorem
- Exceptional set