Exponential representations of injective continuous mappings in radial sets

Magdalena Jastrzębska, Dariusz Partyka

Abstract


By a radial set we understand a non-empty set \(A \subset \mathbb{C} \setminus \{0\}\) such that for every point \(z\in A\) the circle with centre at the origin and passing through \(z\) is included in \(A\). We show in a detailed manner that every continuous and injective function \(F : A \to \mathbb{C} \setminus \{0\}\) can be represented by means of the natural exponential function \(\text{exp}\) and a certain continuous function \(\varPhi : \text{Ei}(A) \to \mathbb{C}\), where \(\text{Ei}(A)\) is the set of all \(z \in \mathbb{C}\) with the property \(\text{exp}(iz) \in A\). The representation is given by \(F(\text{exp}(iz)) = \text{exp}(i\varPhi (z))\) for \(z \in \text{Ei}(A)\). We also touch the problem of the injectivity of \(\varPhi\).

Keywords


Angular parametrization; cuttings of the plane; functional equations; fundamental group of the unit circle; lifted mapping; logarithmic functions of complex variable; quasiconformal mappings

Full Text:

PDF

References


Ahlfors, L. V., Lectures on Quasiconformal Mappings, D. Van Nostrand, Princeton, New Jersey–Toronto–New York–London, 1966.

Ahlfors, L. V., Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd ed., McGraw-Hill, Inc., New York, 1979.

Duren, P., Harmonic Mappings in the Plane, Cambridge University Press, Cambridge,2004.

Eilenberg, S., Transformations continues en circonfernce et la topologie du plan, Fund. Math. 26 (1936), 61–112.

Hatcher, A., Algebraic Topology, Cambridge University Press, Cambridge, 2002.

Kosniowski, C., A First Course in Algebraic Topology, Cambridge University Press, Cambridge, 1980.

Krzyż, J. G., Quasicircles and harmonic measure, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 12 (1987), 19–24.

Kuratowski, K., Introduction to Set Theory and Topology, 2nd English ed., Pergamon Press, Oxford, 2014.

Lehto, O., Virtanen, K. I., Quasiconformal Mappings in the Plane, 2nd ed., Springer, Berlin, 1973.

Mori, A., On quasi-conformality and pseudo-analyticity, Trans. Amer. Math. Soc. 84 (1957), 56–77.

Partyka, D., The generalized Neumann–Poincare operator and its spectrum, Dissertationes Math., vol. 366, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1997.

Rudin, W., Real and Complex Analysis, third ed., McGraw-Hill International Editions, Mathematics Series, McGraw-Hill Book Company, Singapore, 1987.




DOI: http://dx.doi.org/10.17951/a.2021.75.1.37-51
Date of publication: 2021-07-24 12:07:00
Date of submission: 2021-07-21 21:24:37


Statistics


Total abstract view - 614
Downloads (from 2020-06-17) - PDF - 535

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2021 Magdalena Jastrzebska, Dariusz Partyka