Complex Variables and Applications

1 Complex NumbersSums and ProductsBasic Algebraic PropertiesFurther PropertiesModuliComplex ConjugatesExponential FormProducts and Quotients in Exponential FormRoots of Complex NumbersExamplesRegions in the Complex Plane2 Analytic FunctionsFunctions of a Complex VariableMappingsMappings by the Exponential FunctionLimitsTheorems on LimitsLimits Involving the Point at InfinityContinuityDerivativesDifferentiation FormulasCauchy--Riemann EquationsSufficient Conditions for DifferentiabilityPolar CoordinatesAnalytic FunctionsExamplesHarmonic FunctionsUniquely Determined Analytic FunctionsReflection Principle3 Elementary FunctionsThe Exponential FunctionThe Logarithmic FunctionBranches and Derivatives of LogarithmsSome Identities Involving LogarithmsComplex ExponentsTrigonometric FunctionsHyperbolic FunctionsInverse Trigonometric and Hyperbolic Functions4 IntegralsDerivatives of Functions w(t)Definite Integrals of Functions w(t)ContoursContour IntegralsExamplesUpper Bounds for Moduli of Contour IntegralsAntiderivativesExamplesCauchy--Goursat TheoremProof of the TheoremSimply and Multiply Connected DomainsCauchy Integral FormulaDerivatives of Analytic FunctionsLiouville's Theorem and the Fundamental Theorem of AlgebraMaximum Modulus Principle5 SeriesConvergence of SequencesConvergence of SeriesTaylor SeriesExamplesLaurent SeriesExamplesAbsolute and Uniform Convergence of Power SeriesContinuity of Sums of Power SeriesIntegration and Differentiation of Power SeriesUniqueness of Series RepresentationsMultiplication and Division of Power Series6 Residues and PolesResiduesCauchy's Residue TheoremUsing a Single ResidueThe Three Types of Isolated Singular PointsResidues at PolesExamplesZeros of Analytic FunctionsZeros and PolesBehavior of f Near Isolated Singular Points7 Applications of ResiduesEvaluation of Improper IntegralsExampleImproper Integrals from Fourier AnalysisJordan's LemmaIndented PathsAn Indentation Around a Branch PointIntegration Along a Branch CutDefinite Integrals Involving Sines and CosinesArgument PrincipleRouche's TheoremInverse Laplace TransformsExamples8 Mapping by Elementary FunctionsLinear TransformationsThe Transformation w = 1/zMappings by 1/zLinear Fractional TransformationsAn Implicit FormMappings of the Upper Half PlaneThe Transformation w = sin zMappings by z2 and Branches of z1/2Square Roots of PolynomialsRiemann SurfacesSurfaces for Related Functions9 Conformal MappingPreservation of AnglesScale FactorsLocal InversesHarmonic ConjugatesTransformations of Harmonic FunctionsTransformations of Boundary Conditions10 Applications of Conformal MappingSteady TemperaturesSteady Temperatures in a Half PlaneA Related ProblemTemperatures in a QuadrantElectrostatic PotentialPotential in a Cylindrical SpaceTwo-Dimensional Fluid FlowThe Stream FunctionFlows Around a Corner and Around a Cylinder11 The Schwarz--Christoffel TransformationMapping the Real Axis onto a PolygonSchwarz--Christoffel TransformationTriangles and RectanglesDegenerate PolygonsFluid Flow in a Channel Through a SlitFlow in a Channel with an OffsetElectrostatic Potential about an Edge of a Conducting Plate12 Integral Formulas of the Poisson TypePoisson Integral FormulaDirichlet Problem for a DiskRelated Boundary Value ProblemsSchwarz Integral FormulaDirichlet Problem for a Half PlaneNeumann ProblemsAppendixesBibliographyTable of Transformations of RegionsIndex