Functions of One Complex Variable
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 cours...
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| Format: | eBook |
| Language: | English |
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New York, NY
Springer New York
1973, 1973
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| Edition: | 1st ed. 1973 |
| Series: | Graduate Texts in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- §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
- §8. Goursat’s Theorem
- V. Singularities
- §1. Classification ofsingularities
- §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-Lindelöf 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
- 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, Möbius 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