Complex Variables and Applications

8 Rev ed
ISBN: 978-0-07-305194-9
Verlag: McGraw-Hill Education - Europe

Buch, Englisch, 480 Seiten, Format (B × H): 165 mm x 236 mm, Gewicht: 771 g

Complex Variables and Applications, 8e will serve, just as the earlier editions did, as a textbook for an introductory course in the theory and application of functions of a complex variable. This new edition preserves the basic content and style of the earlier editions. The text is designed to develop the theory that is prominent in applications of the subject. You will find a special emphasis given to the application of residues and conformal mappings. To accommodate the different calculus backgrounds of students, footnotes are given with references to other texts that contain proofs and discussions of the more delicate results in advanced calculus. Improvements in the text include extended explanations of theorems, greater detail in arguments, and the separation of topics into their own sections

Autoren/Hrsg.

Brown, James Ward

Brown, James

Churchill, Ruel V.

Churchill, Ruel

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis

Inhaltsverzeichnis

1 Complex NumbersSums and ProductsBasic Algebraic PropertiesFurther PropertiesVectors and ModuliComplex ConjugatesExponential FormProducts and Powers in Exponential FormArguments of Products and Quotients Roots 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 IntegralsSome ExamplesExamples with Branch CutsUpper Bounds for Moduli of Contour IntegralsAntiderivativesProof of the TheoremCauchy–Goursat TheoremProof of the TheoremSimply Connected DomainsMultiply Connected DomainsCauchy Integral FormulaAn Extension of the Cauchy Integral FormulaSome Consequences of the ExtensionLiouville’s Theorem and the Fundamental Theorem of AlgebraMaximum Modulus Principle5 SeriesConvergence of SequencesConvergence of SeriesTaylor SeriesProof of Taylor's TheoremExamplesLaurent SeriesProof of Laurent's TheoremExamplesAbsolute 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 PolesIsolated Singular PointsResiduesCauchy’s Residue TheoremResidue at InfinityThe Three Types of Isolated Singular PointsResidues at PolesExamplesZeros of Analytic FunctionsZeros and PolesBehavior of Functions 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 PrincipleRouché’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

Über Autor(innen)

Churchill, Ruel
RUEL V. CHURCHILL (University of Michigan)