Global mapping properties of analytic functions

Cazacu, Cabiria Andreian; Ghişa, Dorin

doi:10.1142/9789814313179_0001

Cited by 6 publications

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“…In the same year John Derbyshire uses both: the pre-image of the real axis and that of the imaginary axis in his popular book Prime Obsession and declares that they are at the heart of that book. The classification of the components of the pre-image of the real axis by the Riemann Zeta function appears for the first time in (Andreian-Cazacu, Ghisa, 2009), where the strips S k are also introduced and a method is devised of partitioning them into fundamental domains. Later, such a classification has been extended to Dirichlet L-functions and finally to functions defined by general Dirichlet series.…”

Section: Pre-images Of Lines and Circlesmentioning

confidence: 99%

“…Our aim is to show that the complex plane can be divided into a countable number of sets whose interiors are fundamental domains of this function. We have done this previously for the Riemann Zeta function (Andreian-Cazacu, Ghisa, 2009) as well as for Dirichlet L-functions (Ghisa, 2016a). To do the same thing for functions obtained by analytic continuations of general Dirichlet series we made (Ghisa, 2016b) the assumption that (1) satisfies a Riemann type of functional equation.…”

Section: Introductionmentioning

confidence: 99%

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The Geometry of the Mappings by General Dirichlet Series

Ghişa¹

2017

APM

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We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser's theorem. We have also proved that those functions do not possess multiple zeros of order higher than 2 and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof.

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Section: Pre-images Of Lines and Circlesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Geometry of the Mappings by General Dirichlet Series

Ghişa¹

2017

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Fundamental Domains of Gamma and Zeta Functions

Cazacu

Ghişa

2011

International Journal of Mathematics and Mathematical Sciences

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Branched covering Riemann surfaces(ℂ,f)are studied, wherefis the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of cover transformations is revealed. In order to find fundamental domains, preimages of the real axis are taken and a thorough study of their geometry is performed. The technique of simultaneous continuation, introduced by the authors in previous papers, is used for this purpose. Color visualization of the conformal mapping of the complex plane by these functions is used for a better understanding of the theory. A version of this paper containing colored images can be found in arXiv at Andrian Cazacu and Ghisa.

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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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Global mapping properties of analytic functions

Cited by 6 publications

References 0 publications

The Geometry of the Mappings by General Dirichlet Series

The Geometry of the Mappings by General Dirichlet Series

Fundamental Domains of Gamma and Zeta Functions

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

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