Functions of One Complex Variable I

"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. The approach to each topic appears to be carefully thought out both as to mathematical treatment and pedagogical presentation, and the end result is a very satisfactory book." --MATHSCINET

Contenu

I. The Complex Number System.- The real numbers.- The field of complex numbers.- The complex plane.- Polar representation and roots of complex numbers.- Lines and half planes in the complex plane.- The extended plane and its spherical representation.- II. Metric Spaces and the Topology of ?.- Definition and examples of metric spaces.- Connectedness.- Sequences and completeness.- Compactness.- Continuity.- Uniform convergence.- III. Elementary Properties and Examples of Analytic Functions.- Power series.- Analytic functions.- Analytic functions as mapping, Mius transformations.- IV. Complex Integration.- Riemann-Stieltjes integrals.- Power series representation of analytic functions.- Zeros of an analytic function.- The index of a closed curve.- Cauchy's Theorem and Integral Formula.- The homotopic version of Cauchy's Theorem and simple connectivity.- Counting zeros; the Open Mapping Theorem.- Goursat's Theorem.- V. Singularities.- Classification of singularities.- Residues.- The Argument Principle.- VI. The Maximum Modulus Theorem.- The Maximum Principle.- Schwarz's Lemma.- Convex functions and Hadamard's Three Circles Theorem.- Phragm>-Lindel> Theorem.- VII. Compactness and Convergence in ihe Space of Analytic Functions.- The space of continuous functions C(G, ?).- Spaccs of analytic functions.- Spaccs of meromorphic functions.- The Riemann Mapping Theorem.- Weierstrass Factorization Theorem.- Factorization of the sine function.- $7. The gamma function.- The Riemann zeta function.- VIII. Runge's Theorem.- Runge's Theorem.- Simple connectedness.- Mittag-Leffler's Theorem.- IX. Analytic Continuation and Riemann Surfaces.- Schwarz Reflection Principle.- $2. Analytic Continuation Along A Path.- Monodromy Theorem.- Topological Spaces and Neighborhood Systems.- $5. The Sheaf of Germs of Analytic Functions on an Open Set.- $6. Analytic Manifolds.- Covering spaccs.- X. Harmonic Functions.- Basic Properties of harmonic functions.- Harmonic functions on a disk.- Subharmonic and superharmonic functions.- The Dirichlet Problem.- Green's Functions.- XI. Entire Functions.- Jensen's Formula.- The genus and order of an entire function.- Hadamard Factorization Theorem.- XII. The Range of an Analytic Function.- Bloch's Theorem.- The Little Picard Theorem.- Schottky's Theorem.- 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.