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Functions of a Complex Variable
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
The Riemann sphere shows that the point z=∞ is, in some sense, just as valid as the other, finite, points of ℂ¯: both finite and infinite points represent points of the sphere S. The Riemann sphere is often a convenient place to work in situations that require consideration of the point at infinity, or points approaching it. It is possible to show that the stereographic projection maps lines and circles in ℂ to circles in S, and that angles between intersecting curves are preserved.
Complex numbers
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
When c ≠ 0 in Definition 6.1, we notice that f is not defined when z=–dc. It is natural in this case to say that f(–dc)=∞. This simple idea leads to the Riemann sphere; we add a point to the complex plane, called the point at infinity. We discuss this construction in Section 8. First we analyze the geometric properties of linear fractional transformations.
Geometric limit of Julia set of a family of rational functions with odd degree
Published in Dynamical Systems, 2021
A. M. Alves, B. P. Silva e Silva, M. Salarinoghabi
The theory of the iteration of a rational function or transcendental entire function of the complex variable z is widely investigated by many mathematicians since the papers produced, independently, by Fatou [7] and Julia [11], almost 100 years ago. In this theory, we study the sequence of natural iterates, , of a rational function where is the Riemann sphere. In fact, any holomorphic map can be expressed as a rational function, that is, as the quotient of two polynomials. Here we may assume that and have no common roots. The degree d, , of R is then equal to the maximum of the degrees of p and q.
Julia sets for Fibonacci endomorphisms of ℝ2
Published in Dynamical Systems, 2018
S. Bonnot, A. de Carvalho, A. Messaoudi
The field of mathematics which is now called complex dynamics was started by Fatou and Julia in the beginning of the twentieth century: in the late 1910s, they proved a number of results about the iteration of polynomial endomorphisms of the Riemann sphere. The field lay essentially dormant for several decades until in the early 1980s, with the advent of computers, it experienced a vigorous rebirth, became a vibrant area of mathematics which attracted many researchers and eventually produced beautiful results with ties to many other areas of mathematical research (see [9] and [2] for an overview of the area). The study of higher-dimensional complex dynamics started in the late 1980s with the work of Hubbard, Fornaess-Sibony and Bedford-Smillie (see for example [1]). These and other authors studied extensively the complex Hénon family of polynomial diffeomorphisms of .
Efficient iterative scheme for solving non-linear equations with engineering applications
Published in Applied Mathematics in Science and Engineering, 2022
Mudassir Shams, Nasreen Kausar, Praveen Agarwal, Georgia Irina Oros
The dynamical theories of complex dynamics that we apply in this work will now be reviewed (see [28,32]). The orbit of a point is defined as follows given a rational function , where is the Riemann sphere. A point is called fixed point if A periodic point z is a point such that and for In particular, a fixed point is attractor if superattractor if repulsor if and parabolic if Therefore, the super attracting fixed point is also known as the critical point. An attracting point is defined on the basis of attraction, , as the set of starting points whose orbit tends to The scaling theorem enables an appropriate modification of the coordinate to reduce dynamics of iteration of general maps and examines a particular family of iterations of similar maps. As all of the numerical techniques S1–S5 and MM1–MM5 satisfy the scaling theorem and allows dynamics studies.