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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.
PRELIMINARIESThe field of complex numbersThe complex planeMetric spacesMappings and functions. ContinuityTHE CLASSICAL APPROACHOrdinary complex differentiationPreliminaries of the Integral CalculusComplex Integral Theorems AN ALTERNATIVE APPROACHPartial complex differentiationsComplex Green-Gauss Integral TheoremsGeneralized Cauchy Integral FormulaThe classical Cauchy Integral FormulaComparisonLOCAL PROPERTIESExistence of higher order derivatives Local power series representationDistribution of zerosThe Weierstrass Convergence TheoremConnection with plane Potential Theory Complex Integral Theorems revisitedGLOBAL PROPERTIESAnalytic continuationMaximum Modulus PrincipleEntire functionsFundamental Theorem of AlgebraISOLATED SINGULARITIESClassificationLaurent seriesCharacterization by the principal partMeromorphic functionsBehavior at essential singularitiesBehavior at infinityPartial fractions of rational functionsMeromorphic functions on the SphereHOMOTOPYStatement of the problemHomotopic curvesPath independent line integralsSimply connected domainsSolution of first order systemsConjugate solutionsInversion of complex differentiationMorera's TheoremPotentials of vector fields RESIDUE THEORYStatement of the problemWinding numbersThe integration of principal partsResidue TheoremCalculation of residuesAPPLICATIONS OF RESIDUE CALCULUSTotal number of zeros and polesEvaluation of definite integralsSum of certain seriesMAPPING PROPERTIESContinuously differentiable mappingsConformal mappingsExamples of conformal mappingsUnivalent functionsRiemann's Mapping TheoremConstruction of flow linesSPECIAL FUNCTIONSPrescribed principal partsPrescribed zerosInfinite productsWeierstrass productsGamma FunctionThe Riemann Zeta FunctionElliptic FunctionsBOUNDARY VALUE PROBLEMSPreliminariesThe Poisson Integral FormulaCauchy Type IntegralsDesired Holomorphic Functions
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