Selected Topics in the Classical Theory of Functions of a Complex Variable

Selected Topics in the Classical Theory of Functions of a Complex Variable

by Maurice Heins

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Product Details

ISBN-13: 9780486789767
Publisher: Dover Publications
Publication date: 01/14/2015
Series: Dover Books on Mathematics
Pages: 176
Product dimensions: 6.00(w) x 9.00(h) x (d)

About the Author


Maurice Heins taught at the University of Illinois at Champagne–Urbana and at the University of Maryland at College Park.

Table of Contents

Chapter 1 Preliminaries 1

1 Terminology, notation, chordal metric, topology of the extended plane 1

2 The Cauchy-Goursat Theorem 3

3 Meromorphic Functions 9

4 The Mittag-Leffler and Weierstrass Theorems 12

5 Fourier Series and Analytic Functions 15

Chapter 2 Covering Properties of Meromorphic Functions 22

1 Introductory Remarks

2 Continuous Logarithm of a Nonvanishing Continuous 22 Function 22

3 Closed Curve 24

4 Order of a Point with Respect to a Closed Curve 24

5 Argument Principle 25

6 Local Analysis of an Analytic Function 28

7 Lower Semicontinuity of the Valence Function 30

8 Boundary Values of a Meromorphic Function 31

9 Boundary-preserving Maps 34

10 Two Theorems of Landau 36

11 Simple Connectivity 39

12 The Riemann Mapping Theorem 40

13 The Koebe Mapping 41

14 Compactness of a Family of Uniformly Bounded Analytic Functions 42

15 Proof of the Mapping Theorem 43

Chapter 3 The Picard Theorem 45

1 Statement of the Theorem 45

2 The Bloch Theorem 45

3 A Lemma 48

4 Theorem of Schottky 49

5 Proof of the Big Picard Theorem 50

6 A Theorem of Littlewood 52

Chapter 4 Harmonic and Subharmonie Functions 54

1 Introduction 54

A Properties of Harmonic Functions 54

2 Harmonic Functions 54

3 The Poisson Integral 58

B Poisson-Stieltjes Integral. Fatou Theorem 66

4 Positive Harmonic Functions on Δ(0;1) 66

5 The Fatou Theorem 69

C Subharmonic Functions and Applications 71

6 References 71

1 Upper Semicontinuity 71

8 Definition of Subharmonicity 74

9 Properties of Subharmonic Functions 78

10 β≥($$$)(Ahlfors) 83

11 Perron Families 87

D Dirichlet Problem 91

12 The Dirichlet Problem 91

Chapter 5 Applications 96

1 Introduction 96

2 Green's Function 96

3 Classification of Regions 97

4 The Lindelöf Principle 99

5 Iversen's Theorem 102

6 Harmonic Measure 103

7 The Sectorial-limit Theorem 105

8 The Inequality of Miloux-Schmidt 108

9 The Phragmén-Lindelof Theorem 111

10 Wiman's Theorem 115

Chapter 6 The Boundary Behavior of the Rlemann Mapping Function for Simply-connected Jordan Regions 129

1 Some General Observations 129

2 An Example 130

3 The Central Theorem 130

4 The Stronger Form of the Cauchy Integral Theorem 135

Appendix 139

1 Riesz Representation Theorem for C 139

2 Lebesgue's Theorem 141

3 Jordan Curve Theorem. Simple Connectivity 146

Index 157

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