目次

COMPLEX VARIABLESComplex NumbersFinding RootsThe Derivative in the Complex Plane: The Cauchy–Riemann EquationsLine IntegralsCauchy–Goursat TheoremCauchy’s Integral FormulaTaylor and Laurent Expansions and SingularitiesTheory of ResiduesEvaluation of Real Definite IntegralsCauchy’s Principal Value Integral FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONSClassification of Differential EquationsSeparation of VariablesHomogeneous EquationsExact EquationsLinear EquationsGraphical SolutionsNumerical Methods HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONSHomogeneous Linear Equations with Constant CoefficientsSimple Harmonic MotionDamped Harmonic MotionMethod of Undetermined CoefficientsForced Harmonic MotionVariation of ParametersEuler–Cauchy EquationPhase DiagramsNumerical Methods FOURIER SERIESFourier SeriesProperties of Fourier SeriesHalf-Range ExpansionsFourier Series with Phase AnglesComplex Fourier SeriesThe Use of Fourier Series in the Solution of Ordinary Differential EquationsFinite Fourier Series THE FOURIER TRANSFORMFourier TransformsFourier Transforms Containing the Delta FunctionProperties of Fourier TransformsInversion of Fourier TransformsConvolutionSolution of Ordinary Differential Equations by Fourier Transforms THE LAPLACE TRANSFORMDefinition and Elementary PropertiesThe Heaviside Step and Dirac Delta FunctionsSome Useful TheoremsThe Laplace Transform of a Periodic FunctionInversion by Partial Fractions: Heaviside’s Expansion TheoremConvolutionIntegral EquationsSolution of Linear Differential Equations with Constant CoefficientsInversion by Contour Integration THE Z-TRANSFORMThe Relationship of the Z-Transform to the Laplace TransformSome Useful PropertiesInverse Z-TransformsSolution of Difference EquationsStability of Discrete-Time Systems THE HILBERT TRANSFORMDefinitionSome Useful PropertiesAnalytic SignalsCausality: The Kramers–Kronig Relationship THE STURM–LIOUVILLE PROBLEMEigenvalues and EigenfunctionsOrthogonality of EigenfunctionsExpansion in Series of EigenfunctionsA Singular Sturm–Liouville Problem: Legendre’s EquationAnother Singular Sturm–Liouville Problem: Bessel’s EquationFinite Element Method THE WAVE EQUATIONThe Vibrating StringInitial Conditions: Cauchy ProblemSeparation of VariablesD’Alembert’s FormulaThe Laplace Transform MethodNumerical Solution of the Wave Equation THE HEAT EQUATIONDerivation of the Heat EquationInitial and Boundary ConditionsSeparation of VariablesThe Laplace Transform MethodThe Fourier Transform MethodThe Superposition IntegralNumerical Solution of the Heat Equation LAPLACE’S EQUATIONDerivation of Laplace’s EquationBoundary ConditionsSeparation of VariablesThe Solution of Laplace’s Equation on the Upper Half-PlanePoisson’s Equation on a RectangleThe Laplace Transform MethodNumerical Solution of Laplace’s EquationFinite Element Solution of Laplace’s Equation GREEN’S FUNCTIONS What Is a Green’s Function? Ordinary Differential Equations Joint Transform Method Wave EquationHeat Equation Helmholtz’s Equation VECTOR CALCULUSReviewDivergence and CurlLine IntegralsThe Potential FunctionSurface IntegralsGreen’s LemmaStokes’ TheoremDivergence Theorem LINEAR ALGEBRAFundamentals of Linear AlgebraDeterminantsCramer’s RuleRow Echelon Form and Gaussian EliminationEigenvalues and EigenvectorsSystems of Linear Differential EquationsMatrix Exponential PROBABILITYReview of Set Theory Classic Probability Discrete Random Variables Continuous Random Variables Mean and Variance Some Commonly Used Distributions Joint Distributions RANDOM PROCESSES Fundamental ConceptsPower Spectrum Differential Equations Forced by Random Forcing Two-State Markov Chains Birth and Death Processes Poisson Processes Random Walk ANSWERS TO THE ODD-NUMBERED PROBLEMS INDEX