Author | Conway, John B. author |
---|---|
Title | Functions of One Complex Variable [electronic resource] / by John B. Conway |
Imprint | New York, NY : Springer US, 1973 |
Connect to | http://dx.doi.org/10.1007/978-1-4615-9972-2 |
Descript | XIII, 313 p. online resource |
I. The Complex Number System -- ยง1. The real numbers -- ยง2. The field of complex numbers -- ยง3. The complex plane -- ยง4. Polar representation and roots of complex numbers -- ยง5. Lines and half planes in the complex plane -- ยง6. The extended plane and its spherical representation -- II. Metric Spaces and the Topology of C -- ยง1. Definition and examples of metric spaces -- ยง2. Connectedness -- ยง3. Sequences and completeness -- ยง4. Compactness -- ยง5. Continuity -- ยง6. Uniform convergence -- III. Elementary Properties and Examples of Analytic Functions -- ยง1. Power series -- ยง2. Analytic functions -- ยง3. Analytic functions as mappings, Mรถbius transformations -- IV. Complex Integration -- ยง1. Riemann-Stieltjes integrals -- ยง2. Power series representation of analytic functions -- ยง3. Zeros of an analytic function -- ยง4. Cauchyโs Theorem -- ยง5. The index of a closed curve -- ยง6. Cauchyโs Integral Formula -- ยง7. Counting zeros; the Open Mapping Theorem -- ยง8. Goursatโs Theorem -- V. Singularities -- ยง1. Classification of singularities -- ยง2. Residues -- ยง3. The Argument Principle -- VI. The Maximum Modules Theorem -- ยง1. The Maximum Principle -- ยง2. Schwarzโs Lemma -- ยง3. Convex functions and Hadamardโs Three Circles Theorem -- ยง4. Phragmen-Lindelรถf Theorem -- VII. Compactness and Convergence in the Space of Analytic Functions -- ยง1. The space of continuous functions C(G,?) -- ยง2. Spaces of analytic functions -- ยง3. Spaces of meromorphic functions -- ยง4. The Riemann Mapping Theorem -- ยง5. Weierstrass Factorization Theorem -- ยง6. Factorization of the sine function -- ยง7. The gamma function -- ยง8. The Riemann zeta function -- VIII. Rungeโs Theorem -- ยง1. Rungeโs Theorem -- ยง2. Another version of Cauchyโs Theorem -- ยง3. Simple connectedness -- ยง4. Mittag-Lefflerโs Theorem -- IX. Analytic Continuation and Riemann Surfaces -- ยง1. Schwarz Reflection Principle -- ยง2. Analytic Continuation Along A Path -- ยง3. Mondromy Theorem -- ยง4. Topological Spaces and Neighborhood Systems -- ยง5. The Sheaf of Germs of Analytic Functions on an Open Set -- ยง6. Analytic Manifolds -- ยง7. Covering spaces -- X. Harmonic Functions -- ยง1. Basic Properties of harmonic functions -- ยง2. Harmonic functions on a disk -- ยง3. Subharmonic and superharmonic functions -- ยง4. The Dirichlet Problem -- ยง5. Greenโs Functions -- XI. Entire Functions -- ยง1. Jensenโs Formula -- ยง2. The genus and order of an entire function -- ยง3. Hadamard Factorization Theorem -- XII. The Range of an Analytic Function -- ยง1. Blochโs Theorem -- ยง2. The Little Picard Theorem -- ยง3. Schottkyโs Theorem -- ยง4. The Great Picard Theorem -- Appendix: Calculus for Complex Valued Functions on an Interval -- List of Symbols