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A new approach that treats complex analysis in a broad context

This book presents a new approach to one of mathematics' oldest fields. It departs from the tradition of teaching complex analysis as a self-contained subject and, instead, treats the subject as a natural development from calculus. It also shows how complex analysis is used in other areas, exploring connections with calculus, algebra, geometry, topology, and other parts of analysis.

The authors provide the ideal framework for a first-year graduate course in complex analysis--building upon ideas the student is already familiar with and simplifying the transition to advanced topics. The book is also for those using complex numbers and functions in applied fields, including engineering, physics, and other areas.

Function Theory of One Complex Variable

Compares and contrasts complex variable theory with real variable theory Clarifies analytical ideas belonging to complex analysis by separating them from topological issues Discusses some of the current research in the field, including a number of interesting topics not discussed in other textbooks Features many examples as well as 75 illustrations Provides especially thorough exercise sets

Tabla de contenidos provista por Syndetics

• Fundamental Concepts
• Complex Line Integrals
• Applications of the Cauchy Integral
• Meromorphic Functions and Residues
• Zeros of a Holomorphic Function
• Holomorphic Functions as Geometric Mappings
• Harmonic Functions
• Infinite Series and Products
• Applications of Infinite Sums and Products
• Analytic Continuation
• Topology
• Rational Approximation Theory
• Special Classes of Holomorphic Functions
• Hilbert Spaces of Holomorphic Functions, the Bergman Kernel, and Biholomorphic Mappings
• Special Functions and the Prime Number Theorem
• Appendices
• References
• Index

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CHOICE Review

Greene and Krantz have written a thorough introduction to complex analysis at the beginning graduate level. They present complex analysis as an extension of calculus and provide numerous applications in several other areas of mathematics. Both introductory and advanced topics are included. They begin with elementary properties of complex numbers and analytic functions, then turn to complex line integrals with applications, Laurent series, residues, properties of holomorphic and harmonic functions, infinite series and products with applications, and analytic continuation. The authors then analyze many of the more challenging topological aspects of complex analysis. The highlights of the advanced topics are the theorems of Runge and Mergelyan, the Bieberbach conjecture, Hardy spaces and classes, the Bergman kernel, biholomorphic mappings, special functions, and a proof of the prime number theorem. Two appendixes address some useful results from real analysis and a statement and proof of Goursat's theorem. Another strong point is the inclusion of approximately 550 exercises, an unusually large number for a book at this level. In summary, an excellent introduction to complex analysis. Graduate students. D. P. Turner; Faulkner University

Notas de autor provistas por Syndetics

ROBERT E. GREENE is a professor in the Department of Mathematics at the University of California at Los Angeles.

STEVEN G. KRANTZ is a professor in the Department of Mathematics at Washington University in St. Louis, Missouri.