Author | Berenstein, Carlos A. author |
---|---|
Title | Complex Variables [electronic resource] : An Introduction / by Carlos A. Berenstein, Roger Gay |
Imprint | New York, NY : Springer New York, 1991 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-3024-3 |
Descript | XII, 652 p. online resource |
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. Rungeโs Theorem -- 3.2. MittagโLefflerโs 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. Mergelyanโs 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