Introduction to the Theory of Entire Functions

Introduction to the Theory of Entire Functions......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 10
1.1 The Nature of Singular Points......Page 14
1.2 Meromorphic Functions (Definition)......Page 17
1.3 Entire Functions (Definition)......Page 18
1.4–1.8 Maximum and Minimum Modulus......Page 19
1.9 Order of Zeros......Page 26
1.10 Algebraic Entire Functions......Page 27
1.11 Rate of Increase of Maximum Modulus and Definition of Order......Page 28
1.12 The Disjunction of Zeros of a Nonconstant Entire Function......Page 29
1.13–1.14 Fundamental Properties of the Complex Number System: Elementary Theorems on Zeros of Entire Functions......Page 30
1.15 Hadamard's Three-Circle Theorem and Convexity......Page 32
1.16 Infinite Products......Page 35
2.1 Residues......Page 39
2.2–2.3 Expansion of a Meromorphic Function......Page 40
2.4 Expansion of an Entire Function......Page 42
2.6 Hurwitz's Theorem......Page 43
2.7 Picard Theorems for Functions of Finite Order......Page 44
3.1 Inequalities for R {f(z)}......Page 53
3.2 Poisson's Integral Formula......Page 55
3.3 Jensen's Theorem......Page 56
3.4 The Poisson–Jensen Formula......Page 60
3.5 Carleman's Theorem......Page 61
3.6 Schwarz's Lemma......Page 65
3.7 A Theorem of Borel and Carathéodory......Page 66
4.1 Weierstrass Factorization Theorem......Page 69
4.2 Order of an Entire Function......Page 72
4.3 Type of an Entire Function......Page 74
4.4 Growth of f(z) in Unbounded Subdomains of the Plane......Page 75
4.5–4.6 Enumerative Function n(r)......Page 76
4.7 Exponent of Convergence......Page 78
4.8 Genus of a Canonical Product......Page 79
4.9 Hadamard's Factorization Theorem......Page 81
4.10 Order and Exponent of Convergence......Page 84
4.12–4.13 Order and Type of an Entire Function Defined by Power Series......Page 87
4.14 On an Entire Function of an Entire Function (G . Pólya)......Page 93
5.1 The Gamma Function......Page 96
5.2 Analytic Continuation of Γ(z)......Page 101
5.4 Bessel's Function......Page 105
5.5 The Function Fα(z) = exp(-tα) cos zt dt (α > 1)......Page 106
5.6 Order of the Derived Function......Page 108
5.7 Laguerre's Theorem......Page 109
5.8 Convex Sets and Lucas's Theorem......Page 110
5.9–5.10 Mittag-Leffler Theorem......Page 113
6.1 Functions with Real Zeros Only......Page 122
6.2 The Minimum Modulus m(r)......Page 128
6.3 Sequences of Functions......Page 131
6.4 Vitali’s Convergence Theorem......Page 134
6.5 Montel’s Theorem......Page 135
7.1-7.7 Theorems of Phragmén and Lindelöf......Page 137
7.8 The Indicator Function h(ē)......Page 142
7.9 Behavior of m(r)......Page 146
8.1 α-Points of an Entire Function......Page 158
8.2 Borel’s Theorem......Page 159
8.3-8.5 Exceptional-B Values......Page 160
8.6 Exceptional-P Values......Page 161
8.7 Schottky’s Theorem......Page 162
8.9 Landau’s Theorem......Page 168
8.10 Picard’s Second Theorem......Page 170
8.11 Asymptotic Values......Page 172
8.12 Contiguous Paths......Page 174
9.1 Enumerative Functions: N(r,a), m(r,a)......Page 176
9.2 The Nevanlinna Characteristic T(r)......Page 181
9.3 A Bound for m(r, a) on | a | = 1......Page 183
9.4 Order of a Meromorphic Function......Page 187
9.5 Factorization of a Meromorphic Function......Page 188
9.6-9.7 The Ahlfors–Shimizu Characteristic T0(r)......Page 189
Appendix......Page 195
Suggestions for Further Reading......Page 198
Bibliography......Page 199
Index......Page 232
Pure and Applied Mathematics......Page 235