Gamma-Lines: On the Geometry of Real and Complex Functions / Edition 1

by Griogor A. Barsegian

Overview

The history of mathematics is, to a considerable extent, connected with the study of solutions of the equation f(x)=a=const for functions f(x) of one real or complex variable. Therefore, it is surprising that we know very little about solutions of u(x,y)=A=const for functions of two real variables. These two solutions, called level of sets, are very important with regard to applications in physics, biology and economics as they make a map of appropriate processes described by the function u(x,y) for given parameters (x,y). This text explores a concept, Gamma-lines, which generalizes the concept of levels of sets and, at the same time, the concept of a-points. The authors provide a book on Gamma-lines for the broad specialist and show the large range of their field of applications. The general methods proposed in this volume are useful for both physicists and engineers.

Product Details

ISBN-13:	9780367395919
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Series:	Modern Analysis Series
Pages:	192
Product dimensions:	6.00(w) x 9.00(h) x (d)

About the Author

Grigor Barsegian is affiliated to the Institute of Mathematics of the National Academy of Sciences of Armenia and is an academician of the Russian Academy of Natural Sciences. He has published extensively in Russian and is one of the main contributors to the classical value distribution theory in recent years.

Table of Contents

Introduction to the series vii

Preface viii

Introduction 1

1 Tangent variation principle: satellite principles 4

1.1 Modifications of length-area principle 4

1.2 Tangent variation principle 12

1.3 Estimates for collections of Γ-lines 21

1.4 Estimates of lengths of Γ-lines for angular-quasiconformal mappings 29

1.5 Remarks on application of estimates of L(D, Γ) 35

2 Nevanlinna and Ahlfors' theories: additions 40

2.1 Basic concepts and, outcomes of Nevanlinna value distribution theory and Ahlfors' theory of covering surfaces 40

2.2 Geometric deficient values 52

2.3 On some additions to Ahlfors' theory of covering surfaces 64

2.4 Bounds of some integrals 75

3 Γ-lines' approach in the theory of meromorphic functions 82

3.1 Principle of closeness of sufficiently large sets of a-points of meromorphic functions 82

3.2 Integrated version of the principle: connections with known classes of functions 91

4 Distribution of Γ-lines for functions meromorphic in C: Applications 102

4.1 The main results on distribution of F-lines 102

4.2 "Windings" of Γ-lines 110

4.3 Average lengths of Γ-lines along concentric circles and the deficient values 114

4.4 Distribution of Γ-lines and value distribution of modules and real parts of meromorphic functions 117

4.5 The number of Γ-lines crossing rings 119

4.6 Distribution of Gelfond points 120

4.7 Nevanlinna's dream-description of transcendental ramification of Riemann surfaces 124

4.8 The proximity property of a-points of meromorphic functions 134

4.9 A proof of the proximity property of a-points based on investigation of Γ-lines only 140

5 Some applied problems 144

5.1 Γ-lines in physics 144

5.2 On the cross road of value distribution, Γ-lines, free boundary theories and applied mathematics 146

5.3 "Point maps" of physical processes and a-points of general classes of functions 156

5.4 On a-points of some non-holomorphic function 163

References 165

Index 175