Introduction to the Theory of Entire Functions (eBook)

Front Cover 1
Introduction to the Theory of Entire Functions 4
Copyright Page 5
Contents 6
Preface 10
Chapter I. A Study of the Maximum Modulus and Basic Theorems 14
1.1 The Nature of Singular Points 14
1.2 Meromorphic Functions (Definition) 17
1.3 Entire Functions (Definition) 18
1.4–1.8 Maximum and Minimum Modulus 19
1.9 Order of Zeros 26
1.10 Algebraic Entire Functions 27
1.11 Rate of Increase of Maximum Modulus and Definition of Order 28
1.12 The Disjunction of Zeros of a Nonconstant Entire Function 29
1.13–1.14 Fundamental Properties of the Complex Number System: Elementary Theorems on Zeros of Entire Functions 30
1.15 Hadamard's Three-Circle Theorem and Convexity 32
1.16 Infinite Products 35
Chapter II. The Expansion of Functions and Picard Theorems 39
2.1 Residues 39
2.2–2.3 Expansion of a Meromorphic Function 40
2.4 Expansion of an Entire Function 42
2.5 Rouché's Theorem 43
2.6 Hurwitz's Theorem 43
2.7 Picard Theorems for Functions of Finite Order 44
Chapter III. Theorems Concerning the Modulus of a Function and Its Zeros 53
3.1 Inequalities for R {f(z)} 53
3.2 Poisson's Integral Formula 55
3.3 Jensen's Theorem 56
3.4 The Poisson–Jensen Formula 60
3.5 Carleman's Theorem 61
3.6 Schwarz's Lemma 65
3.7 A Theorem of Borel and Carathéodory 66
Chapter IV. Infinite Product Representation: Order and Type 69
4.1 Weierstrass Factorization Theorem 69
4.2 Order of an Entire Function 72
4.3 Type of an Entire Function 74
4.4 Growth of f(z) in Unbounded Subdomains of the Plane 75
4.5–4.6 Enumerative Function n(r) 76
4.7 Exponent of Convergence 78
4.8 Genus of a Canonical Product 79
4.9 Hadamard's Factorization Theorem 81
4.10 Order and Exponent of Convergence 84
4.11 Genus of an Entire Function 87
4.12–4.13 Order and Type of an Entire Function Defined by Power Series 87
4.14 On an Entire Function of an Entire Function (G . Pólya) 93
Chapter V. Standard Functions and Characterization Theorems 96
5.1 The Gamma Function 96
5.2 Analytic Continuation of .(z) 101
5.3 Conjugate Points 105
5.4 Bessel's Function 105
5.5 The Function Fa(z) = exp(-ta) cos zt dt (a > 1)
5.6 Order of the Derived Function 108
5.7 Laguerre's Theorem 109
5.8 Convex Sets and Lucas's Theorem 110
5.9–5.10 Mittag-Leffler Theorem 113
Chapter VI. Functions with Real and/or Negative Zeros: Minimum Modulus I and Sequences of Functions 122
6.1 Functions with Real Zeros Only 122
6.2 The Minimum Modulus m(r) 128
6.3 Sequences of Functions 131
6.4 Vitali’s Convergence Theorem 134
6.5 Montel’s Theorem 135
Chapter VII. Theorems of Phragmén and Lindelöf: Minimum Modulus II 137
7.1-7.7 Theorems of Phragmén and Lindelöf 137
7.8 The Indicator Function h(F) 142
7.9 Behavior of m(r) 146
Chapter VIII. Theorems of Borel, Schottky. Picard, and Landau: Asymptotic Values 158
8.1 a-Points of an Entire Function 158
8.2 Borel’s Theorem 159
8.3-8.5 Exceptional-B Values 160
8.6 Exceptional-P Values 161
8.7 Schottky’s Theorem 162
8.8 Picard’s First Theorem 168
8.9 Landau’s Theorem 168
8.10 Picard’s Second Theorem 170
8.11 Asymptotic Values 172
8.12 Contiguous Paths 174
Chapter IX. Elementary Nevanlinna Theory 176
9.1 Enumerative Functions: N(r,a), m(r,a) 176
9.2 The Nevanlinna Characteristic T(r) 181
9.3 A Bound for m(r, a) on | a | = 1 183
9.4 Order of a Meromorphic Function 187
9.5 Factorization of a Meromorphic Function 188
9.6-9.7 The Ahlfors–Shimizu Characteristic T0(r) 189
Appendix 195
Suggestions for Further Reading 198
Bibliography 199
Index 232
Pure and Applied Mathematics 235