Functions of One Complex Variable

Includes Index

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, Moebius 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>en-Lindel>uf 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.

"This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis.