Like real analysis, complex analysis has generated methods indispensable to mathematics and its applications. Exploring the interactions between these two branches, this book uses the results of real analysis to lay the foundations of complex analysis and presents a unified structure of mathematical analysis as a whole. To set the groundwork and mitigate the difficulties newcomers often experience, An Introduction to Complex Analysis begins with a complete review of concepts and methods from real analysis, such as metric spaces and the Green-Gauss Integral Formula. The approach leads to brief, clear proofs of basic statements - a distinct advantage for those mainly interested in applications. Alternate approaches, such as Fichera's proof of the Goursat Theorem and Estermann's proof of the Cauchy's Integral Theorem, are also presented for comparison. Discussions include holomorphic functions, the Weierstrass Convergence Theorem, analytic continuation, isolated singularities, homotopy, Residue theory, conformal mappings, special functions and boundary value problems.
More than 200 examples and 150 exercises illustrate the subject matter and make this book an ideal text for university courses on complex analysis, while the comprehensive compilation of theories and succinct proofs make this an excellent volume for reference.
PRELIMINARIES The field of complex numbers The complex plane Metric spaces Mappings and functions. Continuity THE CLASSICAL APPROACH Ordinary complex differentiation Preliminaries of the Integral Calculus Complex Integral Theorems AN ALTERNATIVE APPROACH Partial complex differentiations Complex Green-Gauss Integral Theorems Generalized Cauchy Integral Formula The classical Cauchy Integral Formula Comparison LOCAL PROPERTIES Existence of higher order derivatives Local power series representation Distribution of zeros The Weierstrass Convergence Theorem Connection with plane Potential Theory Complex Integral Theorems revisited GLOBAL PROPERTIES Analytic continuation Maximum Modulus Principle Entire functions Fundamental Theorem of Algebra ISOLATED SINGULARITIES Classification Laurent series Characterization by the principal part Meromorphic functions Behavior at essential singularities Behavior at infinity Partial fractions of rational functions Meromorphic functions on the Sphere HOMOTOPY Statement of the problem Homotopic curves Path independent line integrals Simply connected domains Solution of first order systems Conjugate solutions Inversion of complex differentiation Morera's Theorem Potentials of vector fields RESIDUE THEORY Statement of the problem Winding numbers The integration of principal parts Residue Theorem Calculation of residues APPLICATIONS OF RESIDUE CALCULUS Total number of zeros and poles Evaluation of definite integrals Sum of certain series MAPPING PROPERTIES Continuously differentiable mappings Conformal mappings Examples of conformal mappings Univalent functions Riemann's Mapping Theorem Construction of flow lines SPECIAL FUNCTIONS Prescribed principal parts Prescribed zeros Infinite products Weierstrass products Gamma Function The Riemann Zeta Function Elliptic Functions BOUNDARY VALUE PROBLEMS Preliminaries The Poisson Integral Formula Cauchy Type Integrals Desired Holomorphic Functions
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