Functions of one complex variable

著者

• Conway, John B.

書誌事項

Functions of one complex variable

John B. Conway

（Graduate texts in mathematics, 11, 159）

Springer-Verlag, c1973-c1995

• [1]
• [1] : pbk
• 2
• 2 : [softcover]

注記

Bibliography: v. 1, p. 307, v. 2, p. [384]-388

Includes index

内容説明・目次

巻冊次

[1] : pbk ISBN 9780387900629

内容説明

This book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - I) arguments. The actual pre- requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ- entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Mathe- matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc.

目次

• I. The Complex Number System.- 1. The real numbers.- 2. The field of complex numbers.- 3. The complex plane.- 4. Polar representation and roots of complex numbers.- 5. Lines and half planes in the complex plane.- 6. The extended plane and its spherical representation.- II. Metric Spaces and the Topology of C.- 1. Definition and examples of metric spaces.- 2. Connectedness.- 3. Sequences and completeness.- 4. Compactness.- 5. Continuity.- 6. Uniform convergence.- III. Elementary Properties and Examples of Analytic Functions.- 1. Power series.- 2. Analytic functions.- 3. Analytic functions as mappings, Mobius transformations.- IV. Complex Integration.- 1. Riemann-Stieltjes integrals.- 2. Power series representation of analytic functions.- 3. Zeros of an analytic function.- 4. Cauchy's Theorem.- 5. The index of a closed curve.- 6. Cauchy's Integral Formula.- 7. Counting zeros
• the Open Mapping Theorem.- 8. Goursat's Theorem.- V. Singularities.- 1. Classification of singularities.- 2. Residues.- 3. The Argument Principle.- VI. The Maximum Modules Theorem.- 1. The Maximum Principle.- 2. Schwarz's Lemma.- 3. Convex functions and Hadamard's Three Circles Theorem.- 4. Phragmen-Lindelof Theorem.- VII. Compactness and Convergence in the Space of Analytic Functions.- 1. The space of continuous functions C(G,?).- 2. Spaces of analytic functions.- 3. Spaces of meromorphic functions.- 4. The Riemann Mapping Theorem.- 5. Weierstrass Factorization Theorem.- 6. Factorization of the sine function.- 7. The gamma function.- 8. The Riemann zeta function.- VIII. Runge's Theorem.- 1. Runge's Theorem.- 2. Another version of Cauchy's Theorem.- 3. Simple connectedness.- 4. Mittag-Leffler's Theorem.- IX. Analytic Continuation and Riemann Surfaces.- 1. Schwarz Reflection Principle.- 2. Analytic Continuation Along A Path.- 3. Mondromy Theorem.- 4. Topological Spaces and Neighborhood Systems.- 5. The Sheaf of Germs of Analytic Functions on an Open Set.- 6. Analytic Manifolds.- 7. Covering spaces.- X. Harmonic Functions.- 1. Basic Properties of harmonic functions.- 2. Harmonic functions on a disk.- 3. Subharmonic and superharmonic functions.- 4. The Dirichlet Problem.- 5. Green's Functions.- XI. Entire Functions.- 1. Jensen's Formula.- 2. The genus and order of an entire function.- 3. Hadamard Factorization Theorem.- XII. The Range of an Analytic Function.- 1. Bloch's Theorem.- 2. The Little Picard Theorem.- 3. Schottky's Theorem.- 4. The Great Picard Theorem.- Appendix: Calculus for Complex Valued Functions on an Interval.- List of Symbols.

巻冊次

2 ISBN 9780387944609

内容説明

This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.

目次

of Volume II.- 13 Return to Basics.- 1 Regions and Curves.- 2 Derivatives and Other Recollections.- 3 Harmonic Conjugates and Primitives.- 4 Analytic Arcs and the Reflection Principle.- 5 Boundary Values for Bounded Analytic Functions.- 14 Conformal Equivalence for Simply Connected Regions.- 1 Elementary Properties and Examples.- 2 Crosscuts.- 3 Prime Ends.- 4 Impressions of a Prime End.- 5 Boundary Values of Riemann Maps.- 6 The Area Theorem.- 7 Disk Mappings: The Class S.- 15 Conformal Equivalence for Finitely Connected Regions.- 1 Analysis on a Finitely Connected Region.- 2 Conformal Equivalence with an Analytic Jordan Region.- 3 Boundary Values for a Conformal Equivalence Between Finitely Connected Jordan Regions.- 4 Convergence of Univalent Functions.- 5 Conformal Equivalence with a Circularly Slit Annulus.- 6 Conformal Equivalence with a Circularly Slit Disk.- 7 Conformal Equivalence with a Circular Region.- 16 Analytic Covering Maps.- 1 Results for Abstract Covering Spaces.- 2 Analytic Covering Spaces.- 3 The Modular Function.- 4 Applications of the Modular Function.- 5 The Existence of the Universal Analytic Covering Map.- 17 De Branges's Proof of the Bieberbach Conjecture.- 1 Subordination.- 2 Loewner Chains.- 3 Loewner's Differential Equation.- 4 The Milin Conjecture.- 5 Some Special Functions.- 6 The Proof of de Branges's Theorem.- 18 Some Fundamental Concepts from Analysis.- 1 Bergman Spaces of Analytic and Harmonic Functions.- 2 Partitions of Unity.- 3 Convolution in Euclidean Space.- 4 Distributions.- 5 The Cauchy Transform.- 6 An Application: Rational Approximation.- 7 Fourier Series and Cesaro Sums.- 19 Harmonic Functions Redux.- 1 Harmonic Functions on the Disk.- 2 Fatou's Theorem.- 3 Semicontinuous Functions.- 4 Subharmonic Functions.- 5 The Logarithmic Potential.- 6 An Application: Approximation by Harmonic Functions.- 7 The Dirichlet Problem.- 8 Harmonic Majorants.- 9 The Green Function.- 10 Regular Points for the Dirichlet Problem.- 11 The Dirichlet Principle and Sobolev Spaces.- 20 Hardy Spaces on the Disk.- 1 Definitions and Elementary Properties.- 2 The Nevanlinna Class.- 3 Factorization of Functions in the Nevanlinna Class.- 4 The Disk Algebra.- 5 The Invariant Subspaces of Hp.- 6 Szegoe's Theorem.- 21 Potential Theory in the Plane.- 1 Harmonic Measure.- 2 The Sweep of a Measure.- 3 The Robin Constant.- 4 The Green Potential.- 5 Polar Sets.- 6 More on Regular Points.- 7 Logarithmic Capacity: Part 1.- 8 Some Applications and Examples of Logarithmic Capacity.- 9 Removable Singularities for Functions in the Bergman Space.- 10 Logarithmic Capacity: Part 2.- 11 The Transfinite Diameter and Logarithmic Capacity.- 12 The Refinement of a Subharmonic Function.- 13 The Fine Topology.- 14 Wiener's criterion for Regular Points.- References.- List of Symbols.

巻冊次

2 : [softcover] ISBN 9781461269113

内容説明

This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable, the level being gauged for graduate students. It treats several topics in geometric function theory as well as potential theory in the plane, covering in particular: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. A knowledge of integration theory and functional analysis is assumed.

目次

of Volume II.- 13 Return to Basics.- 1 Regions and Curves.- 2 Derivatives and Other Recollections.- 3 Harmonic Conjugates and Primitives.- 4 Analytic Arcs and the Reflection Principle.- 5 Boundary Values for Bounded Analytic Functions.- 14 Conformal Equivalence for Simply Connected Regions.- 1 Elementary Properties and Examples.- 2 Crosscuts.- 3 Prime Ends.- 4 Impressions of a Prime End.- 5 Boundary Values of Riemann Maps.- 6 The Area Theorem.- 7 Disk Mappings: The Class S.- 15 Conformal Equivalence for Finitely Connected Regions.- 1 Analysis on a Finitely Connected Region.- 2 Conformal Equivalence with an Analytic Jordan Region.- 3 Boundary Values for a Conformal Equivalence Between Finitely Connected Jordan Regions.- 4 Convergence of Univalent Functions.- 5 Conformal Equivalence with a Circularly Slit Annulus.- 6 Conformal Equivalence with a Circularly Slit Disk.- 7 Conformal Equivalence with a Circular Region.- 16 Analytic Covering Maps.- 1 Results for Abstract Covering Spaces.- 2 Analytic Covering Spaces.- 3 The Modular Function.- 4 Applications of the Modular Function.- 5 The Existence of the Universal Analytic Covering Map.- 17 De Branges's Proof of the Bieberbach Conjecture.- 1 Subordination.- 2 Loewner Chains.- 3 Loewner's Differential Equation.- 4 The Milin Conjecture.- 5 Some Special Functions.- 6 The Proof of de Branges's Theorem.- 18 Some Fundamental Concepts from Analysis.- 1 Bergman Spaces of Analytic and Harmonic Functions.- 2 Partitions of Unity.- 3 Convolution in Euclidean Space.- 4 Distributions.- 5 The Cauchy Transform.- 6 An Application: Rational Approximation.- 7 Fourier Series and Cesaro Sums.- 19 Harmonic Functions Redux.- 1 Harmonic Functions on the Disk.- 2 Fatou's Theorem.- 3 Semicontinuous Functions.- 4 Subharmonic Functions.- 5 The Logarithmic Potential.- 6 An Application: Approximation by Harmonic Functions.- 7 The Dirichlet Problem.- 8 Harmonic Majorants.- 9 The Green Function.- 10 Regular Points for the Dirichlet Problem.- 11 The Dirichlet Principle and Sobolev Spaces.- 20 Hardy Spaces on the Disk.- 1 Definitions and Elementary Properties.- 2 The Nevanlinna Class.- 3 Factorization of Functions in the Nevanlinna Class.- 4 The Disk Algebra.- 5 The Invariant Subspaces of Hp.- 6 Szegoe's Theorem.- 21 Potential Theory in the Plane.- 1 Harmonic Measure.- 2 The Sweep of a Measure.- 3 The Robin Constant.- 4 The Green Potential.- 5 Polar Sets.- 6 More on Regular Points.- 7 Logarithmic Capacity: Part 1.- 8 Some Applications and Examples of Logarithmic Capacity.- 9 Removable Singularities for Functions in the Bergman Space.- 10 Logarithmic Capacity: Part 2.- 11 The Transfinite Diameter and Logarithmic Capacity.- 12 The Refinement of a Subharmonic Function.- 13 The Fine Topology.- 14 Wiener's criterion for Regular Points.- References.- List of Symbols.

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詳細情報

• NII書誌ID(NCID): BA01200858
• ISBN:
  • 0387900616
  • 0387900624
  • 0387944605
  • 9781461269113
• LCCN: 72096938
• LCCN: 95002331
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: New York
• ページ数/冊数: 2 v.
• 大きさ: 24 cm
• 分類:
  • LCC : QA331
  • DC : 515/.93
  • NDC8 : 413.52
• 件名:
  • LCSH : Functions of complex variables
• 親書誌ID:
  • BA00010074