Ash Complex Variables

1. Auflage 2014
ISBN: 978-1-4832-1619-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 264 Seiten, Web PDF

Complex Variables deals with complex variables and covers topics ranging from Cauchy's theorem to entire functions, families of analytic functions, and the prime number theorem. Major applications of the basic principles, such as residue theory, the Poisson integral, and analytic continuation are given. Comprised of seven chapters, this book begins with an introduction to the basic definitions and concepts in complex variables such as the extended plane, analytic and elementary functions, and Cauchy-Riemann equations. The first chapter defines the integral of a complex function on a path in the complex plane and develops the machinery to prove an elementary version of Cauchy's theorem. Some applications, including the basic properties of power series, are then presented. Subsequent chapters focus on the general Cauchy theorem and its applications; entire functions; families of analytic functions; and the prime number theorem. The geometric intuition underlying the concept of winding number is emphasized. The linear space viewpoint is also discussed, along with analytic number theory, residue theory, and the Poisson integral. This book is intended primarily for students who are just beginning their professional training in mathematics.

Autoren/Hrsg.

Ash, Robert B.

Inhaltsverzeichnis

1;Front Cover;1
2;Complex Variables;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;8
6;Chapter
0. PREREQUISITES;10
6.1;0.1 Basic Definitions;12
6.2;0.2 The Extended Plane;14
6.3;0.3 Analytic Functions;15
6.4;0.4 Cauchy–Riemann
Equations and Applications;16
6.5;0.5 The Elementary Functions;18
6.6;0.6 Logarithms and Roots;20
6.7;Remarks on Notation;21
6.8;References;22
7;Chapter
1. THE ELEMENTARY THEORY ;24
7.1;1.1 Integration on Paths;26
7.2;1.2 Power Series;34
7.3;1.3 Applications;47
7.4;Problems;53
8;Chapter
2. THE GENERAL CAUCHY THEOREM;56
8.1;2.1 Logarithms and Arguments;58
8.2;2.2 The Index of a Point with Respect to a Closed Curve;63
8.3;2.3 Cauchy's Theorem;68
8.4;Problems;77
9;Chapter
3. APPLICATIONS OF THE CAUCHY THEORY;80
9.1;3.1 Singularities;82
9.2;3.2 Residue Theory;92
9.3;3.3 Inverse Functions;103
9.4;3.4 Analytic Mappings of One Disk into Another;107
9.5;3.5 Extension of Cauchy's Theorem and Integral Formula;112
9.6;3.6 The Poisson Integral Formula and Its Applications;114
9.7;3.7 The Jensen and Poisson–Jensen
Formulas;120
9.8;3.8 Analytic Continuation;124
9.9;Problems;135
10;Chapter
4. ENTIRE FUNCTIONS;138
10.1;4.1 Infinite Products;140
10.2;4.2 Canonical Products and the Weierstrass Factorization Theorem;145
10.3;4.3 Order of an Entire Function;151
10.4;4.4 The Hadamard Factorization Theorem;160
10.5;Problems;165
11;Chapter
5. FAMILIES OF ANALYTIC FUNCTIONS;166
11.1;5.1 The Spaces A(U) and C(U);168
11.2;5.2 Riemann Mapping Theorem;178
11.3;5.3 The Homotopic Version of Cauchy's
Theorem;182
11.4;Problems;184
12;Chapter
6. THE PRIME NUMBER THEOREM;188
12.1;6.1 The Gamma Function;190
12.2;6.2 The Riemann Zeta Function;195
12.3;6.3 Proof of the Prime Number Theorem;205
13;Solutions To Problems;212
14;Subject Index;262