Analytic Endomorphisms Of The Riemann Sphere

Complex dynamics is one of the most fascinating subjects of study and research in mathematics. This third volume in the series entitled Non-Invertible Dynamical Systems not only examines topological and analytical properties of the iteration of rational functions on the Riemann sphere (in particular, the Fatou and Julia sets) but also focuses on thermodynamic, ergodic and fractal properties of these functions (notably, equilibrium states, Bowen's formula and Sullivan’s conformal measures). This volume builds on the first two volumes in the series while simultaneously developing some methods and techniques specific to rational functions.

Presenting very recent results in a major research area, this book is addressed to experts and non-experts in the mathematical community alike. The applied issues range from crystallization and dendrite growth to quantum chaos, conveying their significance far into the neighboring disciplines of science.

This volume is a collection of research-and-survey articles by eminent and active workers around the world on the various areas of current research in the theory of analytic functions. Many of these articles emerged essentially from the proceedings of, and various deliberations at, three recent conferences in Japan and Korea: An International Seminar on Current Topics in Univalent Functions and Their Applications which was held in August 1990, in conjunction with the International Congress of Mathematicians at Kyoto, at Kinki University in Osaka; An International Seminar on Univalent Functions, Fractional Calculus, and Their Applications which was held in October 1990 at Fukuoka University; and also the Japan-Korea Symposium on Univalent Functions which was held in January 1991 at Gyeongsang National University in Chinju. Contents:Univalent Logharmonic Extensions onto the Unit Disk or onto an Annulus (Z Abdulhadi & W Hengartner)Hypergeometric Functions and Elliptic Integrals (G D Anderson et al.)A Certain Class of Carathéodory Functions Defined by Conditions on the Unit Circle (J Fuka & Z J Jakubowski)Recent Advances in the Theory of Zero Sets of the Bergman Spaces (E A LeBlanc)Spherical Linear Invariance and Uniform Local Spherical Convexity (W C Ma & D Minda)A Special Differential Subordination and Its Application to Univalency Conditions (S S Miller & P T Mocanu)On the Bernardi Integral Operator (M Nunokawa & D K Thomas)The Quasi-Hadamard Products of Certain Analytic Functions (S Owa)Analytic Solutions of a Class of Briot-Bouquet Differential Equations (S Owa & H M Srivastava)A Certain Class of Generalized Hypergeometric Functions Associated with the Hardy Space of Analytic Functions (H M Srivastava)On the Coefficients of the Univalent Functions of the Nevalinna Classes N1 and N2 (P G Todorov)and other papers Readership: Pure and applied mathematicians. keywords:Analytic Functions;Univalent Functions;Geometric Function Theory;Meromorphic Functions;Multivalent Functions;Hypergeometric Functions;Fractional Calculus;Integral Operators;Starlike Functions;Convex Functions;Close-to-Convex Functions;Special Functions;Convolution Operators;Bergman Spaces;Elliptic Integrals;Briot-Bouquet Differential Equations;Caratheodory Functions;Nevanlinna Classes;Schwarz Functions;Spiral-Like Functions;Dynamical Systems

This volume is a collection of research-and-survey articles by eminent and active workers around the world on the various areas of current research in the theory of analytic functions.Many of these articles emerged essentially from the proceedings of, and various deliberations at, three recent conferences in Japan and Korea: An International Seminar on Current Topics in Univalent Functions and Their Applications which was held in August 1990, in conjunction with the International Congress of Mathematicians at Kyoto, at Kinki University in Osaka; An International Seminar on Univalent Functions, Fractional Calculus, and Their Applications which was held in October 1990 at Fukuoka University; and also the Japan-Korea Symposium on Univalent Functions which was held in January 1991 at Gyeongsang National University in Chinju.

This Workshop in nonlinear dynamics and mathematical physics, organized by the Italian Nuclear Energy Agency (ENEA) in Bologna, is intended to give an updated overview of modern trends in the field of nonlinear dynamics with emphasis on applications to physics, quantum theory, plasma physics and fluid dynamics, optics and electrodynamics, computer simulation, and neural networks.

This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided. We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications. After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc. The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.