WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
Conformal Self-Mappings of the Complex Plane with Arbitrary Number of Fixed Points
Authors: ,
Abstract: There are known conformal self-mappings of the fundamental domains of analytic functions via Möbius
transformations. When two adjacent fundamental domains have a straight line or an arc of a circle as a common
boundary, the Schwarz symmetry principle can be applied for one of those mappings and what we obtain is a
conformal self-mapping of the union of those domains in which each one of the domains is mapped onto itself.
Repeating this operation until the whole plane is exhausted, we obtain a conformal self-mapping of the complex
plane in which every fundamental domain is conformally mapped onto itself. We prove in this paper that this
is true for any analytic function. Since the self-mappings of fundamental domains have each one at least one
fixed point, ultimately, for the self-mapping of the complex plane, we obtain at least as many fixed points as
is the number of fundamental domains. When dealing with a rational function, this number is finite, otherwise
we obtain infinitely many fixed points. Computer experimentation allows the illustration of these concepts for
most of the familiar classes of analytic functions. There are known applications of the Möbius transformations in
physics via the Lorentz group. Relating those application to the present work may contribute to the advancement
of the knowledge in that field.
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Keywords: Conformal mappings, Fundamental domains, Möbius transformations, Dirichlet functions,
Computer experimentation, Steiner net
Pages: 971-979
DOI: 10.37394/23206.2023.22.106