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

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References


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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


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