Function Theory of One Complex Variable
Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples andexercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem,and the Bergman kernel. The authors also treat $Hp$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.
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Complex Line Integrals
Applications of the Cauchy Integral
Meromorphic Functions and Residues
The Zeros of a Holomorphic Function
Holomorphic Functions as Geometric Mappings
Infinite Series and Products
Applications of Infinite Sums and Products
Rational Approximation Theory
Special Classes of Holomorphic Functions
The Prime Number Theorem
The Statement and Proof of Goursats Theorem
analytic continuation apply argument assertion Assume boundary bounded Calculate called Cauchy integral Chapter Choose circle compact complex conclude conformal connected consider constant contains continuous function Corollary course curve defined definition derivative differentiable disc discussion domain element equal equation equivalent example Exercise exists expansion extends f is holomorphic fact Figure finite fixed follows function f given gives harmonic function hence holds holomorphic function idea infinite integral Lemma Let f limit linear mapping multiplicities neighborhood normal Note Notice one-to-one particular plane pole polynomial positive power series precise principle Proof Proposition Prove radius of convergence result satisfies sense sequence simply simply connected singularity space statement subharmonic subset Suppose theorem theory unit disc vanishes write zero