Functions of One Complex Variable I

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 ?. - 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 mapping Möbius transformations. - IV. Complex Integration. - 1. Riemann-Stieltjes integrals. - 2. Power series representation of analytic functions. - 3. Zeros of an analytic function. - 4. The index of a closed curve. - 5. Cauchy's Theorem and Integral Formula. - 6. The homotopic version of Cauchy's Theorem and simple connectivity. - 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 Modulus Theorem. - 1. The Maximum Principle. - 2. Schwarz's Lemma. - 3. Convex functions and Hadamard's Three Circles Theorem. - 4. Phragm>én-Lindel>üf Theorem. - VII. Compactness and Convergence in ihe Space of Analytic Functions. - 1. The space of continuous functions C(G ?). - 2. Spaccs of analytic functions. - 3. Spaccs 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. Simple connectedness. - 3. Mittag-Leffler's Theorem. - IX. Analytic Continuation and Riemann Surfaces. - 1. Schwarz Reflection Principle. - $2. Analytic Continuation Along A Path. - 3. Monodromy Theorem. - 4. Topological Spaces and Neighborhood Systems. - $5. The Sheaf of Germs of Analytic Functions on an Open Set. - $6. Analytic Manifolds. - 7. Covering spaccs. - 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 A: Calculus for Complex Valued Functions on an Interval. - Appendix B: Suggestions for Further Study and Bibliographical Notes. - References. - List of Symbols. Language: English