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by Einar Hille

Edition: REV 05Copyright: 2005

Publisher: Chelsea Publishing Co.

Published: 2005

International: No

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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.

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 Phragmén-Lindelöf principle; 18.2 Dirichlet's problem; Lindelöf'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

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Summary

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.

Table of Contents

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 Phragmén-Lindelöf principle; 18.2 Dirichlet's problem; Lindelöf'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

Publisher Info

Publisher: Chelsea Publishing Co.

Published: 2005

International: No

Published: 2005

International: No

** **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 Phragmén-Lindelöf principle; 18.2 Dirichlet's problem; Lindelöf'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