Complex Variables

An Introduction

Inbunden, Engelska, 1991

Av Carlos A. Berenstein, Roger Gay, Carlos a. Berenstein

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This text gives an overview of the basic properties of holomorphic functions of one complex variable. Topics studied in this overview include a detailed description of differential forms, homotopy theory, and homology theory, as the analytic properties of holomorphic functions, the solvability of the inhomogeneous Cauchy-Riemann equation with emphasis on the notation of compact families, the theory of growth of subharmonic functions, and an introduction to the theory of sheaves, covering spaces and Riemann surfaces. To further illuminate the material, a large number of exercises of differing levels of difficulty have been added.

Produktinformation

Utgivningsdatum1991-05-23
Mått156 x 234 x 37 mm
Vikt1 103 g
FormatInbunden
SpråkEngelska
SerieGraduate Texts in Mathematics
Antal sidor652
Upplaga1991
FörlagSpringer-Verlag New York Inc.
ISBN9780387973494

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1 Topology of the Complex Plane and Holomorphic Functions.- 1.1. Some Linear Algebra and Differential Calculus.- 1.2. Differential Forms on an Open Subset ? of ?.- 1.3. Partitions of Unity.- 1.4, Regular Boundaries.- 1.5. Integration of Differential Forms of Degree 2. The Stokes Formula.- 1.6. Homotopy. Fundamental Group.- 1.7. Integration of Closed 1-Forms Along Continuous Paths.- 1.8. Index of a Loop.- 1.9. Homology.- 1.10. Residues.- 1.11. Holomorphic Functions.- 2 Analytic Properties of Holomorphic Functions.- 2.1. Integral Representation Formulas.- 2.2. The Frechet Space ? (?).- 2,3. Holomorphic Maps.- 2.4. Isolated Singularities and Residues.- 2.5. Residues and the Computation of Definite Integrals.- 2.6. Other Applications of the Residue Theorem.- 2.7, The Area Theorem.- 2.8. Conformal Mappings.- 3 The $$\bar \partial$$-Equation.- 3,1. Runge’s Theorem.- 3.2. Mittag—Leffler’s Theorem.- 3.3. The Weierstrass Theorem.- 3.4. An Interpolation Theorem.- 3.5. Closed Ideals in ? (?).- 3.6. The Operator $$\frac{\partial }{ {\partial \bar z}}$$ Acting on Distributions.- 3.7. Mergelyan’s Theorem.- 3.8. Short Survey of the Theory of Distributions. Their Relation to the Theory of Residues.- 4 Harmonic and Subharmonic Functions.- 4.1. Introduction.- 4.2. A Remark on the Theory of Integration.- 4.3. Harmonic Functions.- 4.4. Subharmonic Functions.- 4.5. Order and Type of Subharmonic Functions in ?.- 4.6. Integral Representations.- 4.7. Green Functions and Harmonic Measure.- 4.8. Smoothness up to the Boundary of Biholomorphic Mappings.- 4.9. Introduction to Potential Theory.- 5 Analytic Continuation and Singularities.- 5.1. Introduction.- 5.2. Elementary Study of Singularities and Dirichlet Series.- 5.3. A Brief Study of the Functions ? and ?.- 5.4.Covering Spaces.- 5.5. Riemann Surfaces.- 5.6. The Sheaf of Germs of Holomorphic Functions.- 5.7. Cocycles.- 5.8. Group Actions and Covering Spaces.- 5.9. Galois Coverings.- 5.10 The Exact Sequence of a Galois Covering.- 5.11. Universal Covering Space.- 5.12. Algebraic Functions, I.- 5.13. Algebraic Functions, II.- 5.14. The Periods of a Differential Form.- 5.15. Linear Differential Equations.- 5.16. The Index of Differential Operators.- References.- Notation and Selected Terminology.