Geodesic
Exponential representations of injective continuous mappings in radial sets
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 1, pp. 37-51.

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By a radial set we understand a non-empty set A ⊂ℂ∖{0} such that for every point z∈ 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 →ℂ∖{0} can be represented by means of the natural exponential function exp and a certain continuous function : Ei(A) →ℂ, where Ei(A) is the set of all z ∈ℂ with the property exp(iz) ∈ A. The representation is given by F(exp(iz)) = exp(i (z)) for z ∈Ei(A). We also touch the problem of the injectivity of .
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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Jastrzębska, Magdalena; Partyka, Dariusz. Exponential representations of injective continuous mappings in radial sets. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 1, pp. 37-51. http://geodesic.mathdoc.fr/item/AUM_2021_75_1_a1/

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