Abstract
In this chapter we present briefly some results of complex analysis which are useful for our theory.
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Notes
- 1.
Harmonicity is preserved by any holomorphic change of coordinates.
- 2.
Or the special cases of this form as a or b tends to infinity so that \(\lambda a\) or \(\lambda b^{-1}\) remains finite, respectively.
- 3.
A curve is Jordan if it is simple closed curve. A domain is Jordan if it’s boundary is Jordan curve.
- 4.
The function h is odd if \(h(-z)=-h(z)\).
- 5.
Remember that a curve is rectifiable, if its arc length is finite, and also that \(\int \mathrm {d}^2 z\) denotes the integral with respect to the Lebesgue area measure and \(\int |\mathrm {d}z|\) is the integral w.r.t. the arc length measure.
References
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Duren, P.L.: Univalent Functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259. Springer-Verlag, New York (1983)
Garnett, J.B., Marshall, D.E.: Harmonic Measure, New Mathematical Monographs, vol. 2. Cambridge University Press, Cambridge (2008). Reprint of the 2005 original
Greene, R.E., Krantz, S.G.: Function theory of one complex variable, Graduate Studies in Mathematics, vol. 40, 3rd edn. American Mathematical Society, Providence, RI (2006). https://doi.org/10.1090/gsm/040
Pommerenke, C.: Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer-Verlag, Berlin (1992). https://doi.org/10.1007/978-3-662-02770-7
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)
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Kemppainen, A. (2017). Introduction to Conformal Mappings. In: Schramm–Loewner Evolution. SpringerBriefs in Mathematical Physics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-65329-7_3
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