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Full Description
This text emphasizes the special functions that are used in complex analysis. Starting with the algebraic system of complex numbers, it offers an entry-level course on complex analysis of one variable. It presents the study of analytic functions, conformal mapping, analysis of singularities, and the computation of various integrals. The final three chapters introduce more advanced topics and applications. The book provides examples of applications to various physical problems and explains how to use Mathematica (R), Maple (TM), and MATLAB (R).
Contents
Complex number systemAlgebraic structureTriangle inequality and polar formTopological propertiesStereographic projectionAnalytic functionsFunctions of a complex variableLimit and continuityDifferentiability and analytic functionsCauchy-Riemann equationsApplications to problems in potential flowHarmonic functionsPower series and elementary functionsSequences and seriesUniform convergencePower seriesElementary functionsMultiple valued functionsComplex integrationRectifiable arcs and contoursLine integralCauchy fundamental theorem and its implicationsCauchy theorem in various formsCauchy integral formula and its consequencesMorera theoremFundamental theorem of AlgebraSpecial functionsGamma functionsHypergeometric functionsRiemann zeta functionsConformal mappingMaximum modulus principleSchwarz lemmaSubordinationDefinition of conformal mappingBilinear transformationMappings related to discs and half-planesSpecific transformations such as 1/z, exp(z), sinz and zA2Series representation and singularitiesTaylor series and its consequencesLaurent seriesZeros and Poles of a functionClassification of residuesMeromorphic functionsResidue theorem and evaluation of integralsResidue theoremArgument principle and Rouche's theoremContour integration and applicationsImproper integrals and evaluation of a real integralIntegrals involving sines and cosinesIntegration through branch cutApplications of conformal MappingApplication to steady temperatureApplication to electrostatic potentialApplications to fluid flowSchwarz-Christoffel transformationDirichlet problemNeumann problemHarmonic functionsHarmonic functions and mean value propertyMaximum principlePoisson integral formulaHarnack PrincipleSubharmonic and superharmonic functionsTopological properties of the class of analytic functionsNormal familiesArzela Ascoli theoremRiemann Mapping theoremWeierstrass Factorization theoremUnivalent function theorySubclasses of univalent functionsAnalytic continuationSchwarz reflection principleEntire functions and canonical productsJensen's formulaGenus and order of an entire functionHadamard theoremInfinite productsCanonical product and Weirstrass theoremMittag Leffler's theoremElliptic functionsDoubly periodic functionModular transformationProperties of elliptic functionsWeierstrass elliptic functionsRelated elliptic functionsAppendices
