Description
Book SynopsisEmphasizes the conceptual and historical continuity of analytic function theory. This work covers topics including elliptic functions, entire and meromorphic functions, as well as conformal mapping. It features chapters on majorization and on functions holomorphic in a half-plane.
Table of ContentsVolume II:; 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; 11. Singularities and representation of analytic functions:; 11.1 Holomorphy-preserving transformations: I. Integral operators; 11.2 Holomorphy-preserving transformations: II. 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 noncontinuable power series; 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; 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; 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; 15. Normal families:; 15.1 Schwarz's lemma and hyperbolic measure; 15.2 Normal families; 15.3 Induced convergence; 15.4 Applications; 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; 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; 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; 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 Bibliography Index.
