This famous work is a textbook that emphasizes the conceptual and historical continuity of analytic function theory. The second volume broadens from a textbook to a textbook-treatise, covering the ""canonical"" topics (including elliptic functions, entire and meromorphic functions, as well as conformal mapping, etc.) and other topics nearer the expanding frontier of analytic function theory. In the latter category are the chapters on majorization and on functions holomorphic in a half-plane.
Front Cover Foreword Contents Volume I Volume II Symbols 10 ANALYTIC CONTINUATION 10.1. Introduction. 10.2. Rearrangements of power series. 10.3. Analytic functions. 10.4. Singularities. 10.5. Borel monogenic functions. 10.6. Multivalued functions and Riemann surfaces. 10.7. Law of permanence of functional equations. COLLATERAL READING 11 SINGULARITIES AND REPRESENTATION OF ANALYTIC FUNCTIONS 11.1. Holomorphy-preserving transformations: I. Integral operators. 11.2. Holomorphy-preserving transformations: Il. Differential operators. 11.3. Power series with analytic coefficients. 11.4. Analytic continuation in a star. 11.5. Polynomial series. 11.6. Composition theorems. 11.7. Gap theorems and nonoontinuable power series. COLLATERAL READING 12 ALGEBRAIC FUNCTIONS 12.1. Local properties. 12.2. Critical points. 12.3. Newton' s diagram. 12.4. Riemann surfaces some concepts of algebraic geometry. 12.5. Rational functions on the surface and Abelian integrals. COLLATERAL READING 13 ELLIPTIC FUNCTIONS 13.1. Doubly-periodic functions. 13.2. The functions of Weierstrass. 13.3. Some further properties of elliptic functions. 13.4. On the functions of Jacobi. 13.5. The theta functions. 13.6. Modular functions. COLLATERAL READING 14 ENTIRE AND MEROMORPHIC FUNCTIONS 14.1. Order relations for entire functions. 14.2. Entire functions of finite order. 14.3. Functions with real zeros. 14.4. Characteristic functions. 14.5. Picard' s and Landau' s theorems. 14.6. The second fundamental theorem. 14.7. Defect relations. COLLATERAL READING 15 NORMAL FAMILIES 15.1. Schwarz' s lemma and hyperbolic measure. 15.2. Normal families. 15.3. Induced convergence. 15.4. Applications. COLLATERAL READING 16 LEMNISCATES 16.1. Chebichev polynomials. 16.2. The transfinite diameter. 16.3. Additive set functions Radon-Stieltjes integrals. 16.4. Logarithmic capacity. 16.5. Green' s function Hilbert' s theorem. 16.6. Runge' s theorem. 16. 7. Overconvergence. COLLATERAL READING 17 CONFORMAL MAPPING 17.1. Riemann' s mapping theorem. 17.2. The kernel function. 17.3. Fekete polynomials and the exterior mapping problem. 17.4. Univalent functions. 17.5. The boundary problem. 17.6. Special mappings. 17.7. The theorem of Bloch. COLLATERAL READING 18 MAJORIZATION 18.1. The Phragmen-Lindelof Principle. 18.2. Dirichlet' s problem Lindelof' s principle. 18.3. Harmonic measure. 18.4. The Nevanlinna-Ahlfors-Heins theorems. 18.5. Subordination. COLLATERAL READING 19 FUNCTIONS HOLOMORPHIC IN A HALF-PLANE 19.1. The Hardy-Lebesgue classes. 19.2. Bounded functions. 19.3. Growth-measuring functions. 19.4. Remarks on Laplace-Stieltjes integrals. COLLATERAL READING Bibliography Index Back Cover
