Complex analysis: an introduction to the theory of analytic functions of one complex variableA standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material. Difficult points have been clarified, the book has been reviewed for accuracy, and notations and terminology have been modernized. Chapter 2, Complex Functions, features a brief section on the change of length and area under conformal mapping, and much of Chapter 8, GlobalAnalytic Functions, has been rewritten in order to introduce readers to the terminology of germs and sheaves while still emphasizing that classical concepts are the backbone of the theory. Chapter 4, Complex Integration, now includes a new and simpler proof of the general form of Cauchy's theorem. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals. 
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Review: Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable (International Series in Pure & Applied Mathematics)
User Review  Deni Lennon  GoodreadsUsed it during my complex analysis class. Great choice, but you still have to search additional information. Read full review
Review: Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable (International Series in Pure & Applied Mathematics)
User Review  GoodreadsUsed it during my complex analysis class. Great choice, but you still have to search additional information. Read full review
Contents
The Geometric Representation of Complex Numbers  12 
COMPLEX FUNCTIONS  21 
Elementary Theory of Power Series  33 
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algebraic analytic continuation analytic function angle arbitrary assume bounded branch Cauchy's theorem Chap choose closed curve coefficients compact set complement complex numbers conclude condition conformal mapping conjugate consider constant corresponding defined definition denote derivative differential equation end points entire function equal exists finite number follows formula func function elements function f(z geometric global analytic function harmonic function hence homotopic identically zero imaginary inequality infinite initial germ inverse lemma limit line segment linear transformation meromorphic function metric space multiple neighborhood notation obtain open sets poles polynomial positive power series proof prove radius of convergence real axis rectangle region ft residue Riemann surface roots satisfies sequence simple simply connected simply connected region singlevalued singularity solution subharmonic subset Suppose tion topological totally bounded uniform convergence uniformly upper half plane vanishes variable whole plane