Global mapping properties of rational functions

Ballantine, Christina; Ghişa, Dorin

doi:10.1142/9789814313179_0002

Cited by 3 publications

(3 citation statements)

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“…They are implemented in Mathematica and some affirmations about general Dirichlet functions are illustrated by using Dirichlet L-functions. However, the interest in more general functions is obvious and we have recently devoted to them a lot of publications (see [2]- [15]). An account of recent advances in this field can be found in [8].…”

Section: General Properties Of Dirichlet Functionsmentioning

confidence: 99%

“…for the extended function when it exists and we call it Dirichlet function. Following Speiser [16], who studied the Riemann Zeta function, we have used in [2]- [15] the pre-image of the real axis by We have proved (see for example [8] In the case of a strip , 0 when is the case, as in Figure 2). These are strips unbounded to the right and to the left when boundaries of fundamental domains bounded to the right.…”

Section: General Properties Of Dirichlet Functionsmentioning

confidence: 99%

“…We make reference to [1] for elementary knowledge in complex analysis used below. It is known (see [2]) that for every rational function ( ) R z of degree n the complex plane can be partitioned into n sets whose interior are fundamental domains of ( ) R z , i.e. they are mapped conformally (hence bijectively) by ( ) R z onto the whole complex plane with some slits.…”

Section: Introductionmentioning

confidence: 99%

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The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

Ghişa¹

2020

APM

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The Cauchy integral formula expresses the value of a function () f z , which is analytic in a simply connected domain D, at any point 0 z interior to a simple closed contour C situated in D in terms of the values of () 0 f z z z − on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of () f z. At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of () f z is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle z r = and of some circles 1 z a r − = for which the formula is valid.

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Section: General Properties Of Dirichlet Functionsmentioning

confidence: 99%

Section: General Properties Of Dirichlet Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

Ghişa¹

2020

APM

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Conformal Self-Mappings of the Complex Plane with Arbitrary Number of Fixed Points

Albişoru,

Ghişa

2023

Wseas Transactions on Mathematics

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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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Conformal Self Mappings of the Fundamental Domains of Analytic Functions and Computer Experimentation

Albişoru,

Ghişa

2023

Wseas Transactions on Mathematics

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Conformal self mappings of a given domain of the complex plane can be obtained by using the Riemann Mapping Theorem in the following way. Two different conformal mappings φ and ψ of that domain onto one of the standard domains: the unit disc, the complex plane or the Riemann sphere are taken and then ψ −1 ◦ φ is what we are looking for. Yet, this is just a theoretical construction, since the Riemann Mapping Theorem does not offer any concrete expression of those functions. The Möbius transformations are concrete, but they can be used only for particular circular domains. We are proving in this paper that conformal self mappings of any fundamental domain of an arbitrary analytic function can be obtained via Möbius transformations as long as we allow that domain to have slits. Moreover, those mappings enjoy group properties. This is a totally new topic. Although fundamental domains of some elementary functions are well known, the existence of such domains for arbitrary analytic functions has been proved only in our previous publications mentioned in the References section. No other publication exists on this topic and the reference list is complete. We deal here with conformal self mappings of fundamental domains in its whole generality and present sustaining illustrations. Those related to the case of Dirichlet functions represent a real achievement. Computer experimentation with these mappings are made for the most familiar analytic functions.

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Global mapping properties of rational functions

Abstract: We investigate the fundamental domains of rational functions and provide visualizations for relevant examples. The fundamental domains give a thorough understanding of the global properties of the functions studied.

Cited by 3 publications

References 1 publication

The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

Conformal Self-Mappings of the Complex Plane with Arbitrary Number of Fixed Points

Conformal Self Mappings of the Fundamental Domains of Analytic Functions and Computer Experimentation

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