Introduction to the theory of entire functions, Volume 56 (Pure and Applied Mathematics)

Introduction to the Theory of Entire Functions Copyright Page Contents Preface Chapter I. A Study of the Maximum Modulus and Basic Theorems 1.1 The Nature of Singular Points 1.2 Meromorphic Functions (Definition) 1.3 Entire Functions (Definition) 1.4–1.8 Maximum and Minimum Modulus 1.9 Order of Zeros 1.10 Algebraic Entire Functions 1.11 Rate of Increase of Maximum Modulus and Definition of Order 1.12 The Disjunction of Zeros of a Nonconstant Entire Function 1.13–1.14 Fundamental Properties of the Complex Number System: Elementary Theorems on Zeros of Entire Functions 1.15 Hadamard's Three-Circle Theorem and Convexity 1.16 Infinite Products Chapter II. The Expansion of Functions and Picard Theorems 2.1 Residues 2.2–2.3 Expansion of a Meromorphic Function 2.4 Expansion of an Entire Function 2.5 Rouché's Theorem 2.6 Hurwitz's Theorem 2.7 Picard Theorems for Functions of Finite Order Chapter III. Theorems Concerning the Modulus of a Function and Its Zeros 3.1 Inequalities for R {f(z)} 3.2 Poisson's Integral Formula 3.3 Jensen's Theorem 3.4 The Poisson–Jensen Formula 3.5 Carleman's Theorem 3.6 Schwarz's Lemma 3.7 A Theorem of Borel and Carathéodory Chapter IV. Infinite Product Representation: Order and Type 4.1 Weierstrass Factorization Theorem 4.2 Order of an Entire Function 4.3 Type of an Entire Function 4.4 Growth of f(z) in Unbounded Subdomains of the Plane 4.5–4.6 Enumerative Function n(r) 4.7 Exponent of Convergence 4.8 Genus of a Canonical Product 4.9 Hadamard's Factorization Theorem 4.10 Order and Exponent of Convergence 4.11 Genus of an Entire Function 4.12–4.13 Order and Type of an Entire Function Defined by Power Series 4.14 On an Entire Function of an Entire Function (G . Pólya) Chapter V. Standard Functions and Characterization Theorems 5.1 The Gamma Function 5.2 Analytic Continuation of Γ(z) 5.3 Conjugate Points 5.4 Bessel's Function 5.5 The Function Fα(z) = exp(-tα) cos zt dt (α > 1) 5.6 Order of the Derived Function 5.7 Laguerre's Theorem 5.8 Convex Sets and Lucas's Theorem 5.9–5.10 Mittag-Leffler Theorem Chapter VI. Functions with Real and/or Negative Zeros: Minimum Modulus I and Sequences of Functions 6.1 Functions with Real Zeros Only 6.2 The Minimum Modulus m(r) 6.3 Sequences of Functions 6.4 Vitali’s Convergence Theorem 6.5 Montel’s Theorem Chapter VII. Theorems of Phragmén and Lindelöf: Minimum Modulus II 7.1-7.7 Theorems of Phragmén and Lindelöf 7.8 The Indicator Function h(ē) 7.9 Behavior of m(r) Chapter VIII. Theorems of Borel, Schottky. Picard, and Landau: Asymptotic Values 8.1 α-Points of an Entire Function 8.2 Borel’s Theorem 8.3-8.5 Exceptional-B Values 8.6 Exceptional-P Values 8.7 Schottky’s Theorem 8.8 Picard’s First Theorem 8.9 Landau’s Theorem 8.10 Picard’s Second Theorem 8.11 Asymptotic Values 8.12 Contiguous Paths Chapter IX. Elementary Nevanlinna Theory 9.1 Enumerative Functions: N(r,a), m(r,a) 9.2 The Nevanlinna Characteristic T(r) 9.3 A Bound for m(r, a) on | a | = 1 9.4 Order of a Meromorphic Function 9.5 Factorization of a Meromorphic Function 9.6-9.7 The Ahlfors–Shimizu Characteristic T0(r) Appendix Suggestions for Further Reading Bibliography Index Pure and Applied Mathematics The purpose of this book is to acquaint the reader with some of the basic analysis and theorems central to a study of functions analytic in the entire finite complex plane. This will display some of the intrinsic beauty of the subject, and array in a sequential form theorems central to the study of entire functions, and are basic to an understanding of the geometry of zeros, the growth and behavior of the maximum modulus and the minimum modulus