Extended Complex Plane, Linear-Fractional Transformations, Meromorphic Functions

2020 ◽  
pp. 53-57
Author(s):  
John Mac Sheridan Nerney
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Linear Fractional Transformations ◽  
Extended Complex ◽  
Extended Complex Plane

scholarly journals Discrete free products of two complex cyclic matrix groups

1979 ◽  
Vol 20 (1) ◽  
pp. 69-80 ◽  
Author(s):  
Ronald J. Evans
Keyword(s):  
Complex Plane ◽  
Imaginary Axis ◽  
Free Products ◽  
Matrix Groups ◽  
Linear Fractional Transformations ◽  
Real Trace ◽  
Extended Complex ◽  
Image Position ◽  
The Right ◽  
Extended Complex Plane

All 2-by-2 matrices in this paper are to be viewed as linear fractional transformations on the extended complex plane ℂ*. Let L+ and L− be the open half-planes to the right and left, respectively, of the extended imaginary axis L. Let Λ be the set of complex 2-by-2 matrices A with real trace and determinant ±1 such that A(L+) ⊂L−. Let Ω = Ω1 ∪ Ω2 ∪ Ω3 ∪ Ω4, Whereand

scholarly journals Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Abhijit Banerjee ◽  
Saikat Bhattacharyya
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Shift Operator ◽  
Future Research ◽  
Weighted Sharing ◽  
Shift Operators ◽  
Extended Complex ◽  
Open Question ◽  
Extended Complex Plane

AbstractIn the paper, we introduce a new notion of reduced linear c-shift operator $L _{c}^{r}\,f$Lcrf, and with the aid of this new operator, we study the uniqueness of meromorphic functions $f(z)$f(z) and $L_{c}^{r}\,f$Lcrf sharing two or more values in the extended complex plane. The results obtained in the paper significantly improve a number of existing results. Further, using the notion of weighted sharing of sets, we deal the same problem. We exhibit a handful number of examples to justify certain statements relevant to the content of the paper. We are also able to determine the form of the function that coincides with its reduced linear c-shift operator. At the end of the paper, we pose an open question for future research.

scholarly journals A Construction of Meromorphic Functions with Prescribed Asymptotic Behavior

1971 ◽  
Vol 41 ◽  
pp. 75-87 ◽  
Author(s):  
J.L. Stebbins
Keyword(s):  
Asymptotic Behavior ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
The Other ◽  
Asymptotic Values ◽  
Other Hand ◽  
Extended Complex ◽  
The Given ◽  
Prescribed Asymptotic Behavior ◽  
Extended Complex Plane

Although there are several constructions of meromorphic functions with prescribed asymptotic sets [e.g., 5,6], it is usually difficult to determine or prescribe the nature of the asymptotic paths used in these constructions. On the other hand, there are several other constructions of meromorphic functions with prescribed asymptotic paths [e.g., 1, 10, 12], but the extent of the asymptotic values for these functions cannot always be restricted to the values approached along the given paths. Gross [3] has accomplished both results by prescribing paths for every value in the extended complex plane.

Groups Generated by two Parabolic Linear Fractional Transformations

1969 ◽  
Vol 21 ◽  
pp. 1388-1403 ◽  
Author(s):  
R. C. Lyndon ◽  
J. L. Ullman
Keyword(s):  
Complex Plane ◽  
Trivial Case ◽  
Linear Fractional Transformations ◽  
Extended Complex ◽  
Fractional Group ◽  
Image Position ◽  
Extended Complex Plane

We are interested in the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B, and, particularly, in the question of when such a group G is free. We shall, as usual, represent elements of G by matrices with determinant 1, which are determined up to change of sign. Two such groups G will be conjugate in the full linear fractional group, and hence isomorphic, provided they have, up to a change of sign, the same value of the invariant τ = Trace(AB) – 2. We put aside the trivial case that τ = 0, where G is abelian. In the study of these groups, two normalizations have proved convenient. Sanov (17) and Brenner (3) took the generators in the formwhile Chang, Jennings, and Ree (4) took them in the form

scholarly journals On a Uniqueness Theorem

1969 ◽  
Vol 35 ◽  
pp. 151-157 ◽  
Author(s):  
V. I. Gavrilov
Keyword(s):  
Unit Circle ◽  
Uniqueness Theorem ◽  
Unit Disk ◽  
Complex Plane ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Extended Complex Plane

1. Let D be the open unit disk and r be the unit circle in the complex plane, and denote by Q the extended complex plane or the Rie-mann sphere.

scholarly journals Extended Complex Plane and Riemann Sphere

2020 ◽  
Vol 08 (04) ◽  
pp. 44-51
Author(s):  
Egahi M. ◽  
Agbata B.C. ◽  
Ogwuche O.I. ◽  
Soomiyol M. C
Keyword(s):  
Complex Plane ◽  
Riemann Sphere ◽  
Extended Complex ◽  
Extended Complex Plane

scholarly journals Types of complete Infinitely sheeted Planes

2004 ◽  
Vol 176 ◽  
pp. 181-195 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Complex Plane ◽  
Parameter Family ◽  
Real Numbers ◽  
Extended Complex ◽  
Extended Complex Plane

AbstractWe will answer negatively to the question whether the completeness of infinitely sheeted covering surfaces of the extended complex plane have anything to do with their types being parabolic or hyperbolic. This will be accomplished by giving a one parameter family {W[α]: α ∈ A} of complete infinitely sheeted planes W[α] depending on the parameter set A of sequences α = (an)n>1 of real numbers 0 < an ≤ 1/2 (n ≥ 1) such that W[α] is parabolic for ‘small’ α’s and hyperbolic for ‘large’ α’s.

scholarly journals Quasiconformal extensions for some geometric subclasses of univalent functions

1984 ◽  
Vol 7 (1) ◽  
pp. 187-195 ◽  
Author(s):  
Johnny E. Brown
Keyword(s):  
Recent Result ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Plane ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Sufficiency Condition ◽  
Extended Complex ◽  
Extended Complex Plane

LetSdenote the set of all functionsfwhich are analytic and univalent in the unit diskDnormalized so thatf(z)=z+a2z2+…. LetS∗andCbe those functionsfinSfor whichf(D)is starlike and convex, respectively. For0≤k<1, letSkdenote the subclass of functions inSwhich admit(1+k)/(1−k)-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a functionfbelongs toSk⋂S∗orSk⋂C. Functions whose derivatives lie in a half-plane are also considered and a Noshiro-Warschawski-Wolff type sufficiency condition is given to determine which of these functions belong toSk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.

QUASICONFORMAL HARMONIC MAPPINGS BETWEEN DOMAINS CONTAINING INFINITY

2020 ◽  
Vol 102 (1) ◽  
pp. 109-117
Author(s):  
DAVID KALAJ
Keyword(s):  
Quasiconformal Mapping ◽  
Complex Plane ◽  
Riemannian Metrics ◽  
Harmonic Mappings ◽  
Lipschitz Continuous ◽  
Extended Complex ◽  
Extended Complex Plane

Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$. We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.

Ergodic properties of random iterations of analytic functions

1999 ◽  
Vol 19 (6) ◽  
pp. 1379-1388
Author(s):  
AMIRAN AMBROLADZE
Keyword(s):  
Complex Plane ◽  
Analytic Functions ◽  
Entire Functions ◽  
Iterated Function System ◽  
Ergodic Property ◽  
Function System ◽  
Ergodic Properties ◽  
Iterated Function ◽  
Extended Complex ◽  
Extended Complex Plane

It is a known fact that an iterated function system (IFS) of entire functions is not necessarily ergodic. In this paper we show that if an IFS of analytic functions is defined in a domain whose boundary contains more than two points (in the extended complex plane) then the system possesses an ergodic property.

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