Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. It will primarily be used by students with a background in ordinary differential equations and advanced calculus. There are two main objectives of this text. The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations.
Auflage
Sprache
Verlagsort
Verlagsgruppe
McGraw-Hill Education - Europe
Zielgruppe
Für höhere Schule und Studium
Maße
Höhe: 239 mm
Breite: 170 mm
Dicke: 25 mm
Gewicht
ISBN-13
978-0-07-125917-0 (9780071259170)
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Schweitzer Klassifikation
RUEL V. CHURCHILL (University of Michigan)
Preface1 Fourier SeriesPiecewise Continuous FunctionsFourier Cosine SeriesExamplesFourier Sine SeriesExamplesFourier SeriesExamplesAdaptations to Other Intervals2 Convergence of Fourier SeriesOne-Sided DerivativesA Property of Fourier CoefficientsTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryConvergence on Other IntervalsA LemmaAbsolute and Uniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier Series3 Partial Differential Equations of PhysicsLinear Boundary Value ProblemsOne-Dimensional Heat EquationRelated EquationsLaplacian in Cylindrical and Spherical CoordinatesDerivationsBoundary ConditionsA Vibrating StringVibrations of Bars and MembranesGeneral Solution of the Wave EquationTypes of Equations and Boundary Equations4 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA Temperature ProblemA Vibrating String ProblemHistorical Development5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatSteady Temperatures in a Rectangular PlateCylindrical CoordinatesA String with Prescribed Initial ConditionsResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Fourier Integrals and ApplicationsThe Fourier Integral FormulaDirichlet's IntegralTwo LemmasA Fourier Integral TheoremThe Cosine and Sine IntegralsMore on Superposition of SolutionsTemperatures in a Semi-Infinite SolidTemperatures in an Unlimited Medium7 Orthonormal SetsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and Parseval's EquationApplications to Fourier Series8 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsA Temperature Problem in Rectangular CoordinatesAnother ProblemOther CoordinatesA Modification of the MethodAnother ModificationA Vertically Hung Elastic Bar9 Bessel Functions and ApplicationsBessel Functions Jn(x)General Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormSome Consequences of the Integral FormsThe Zeros of Jn(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of the TheoremsThe Orthonormal FunctionsFourier-Bessel SeriesExamplesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane10 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos ?)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere11 Verification of Solutions and UniquenessAbel's Test for Uniform ConvergenceVerification of Solution of Temperature ProblemUniqueness of Solutions of the Heat EquationVerification of Solution of Vibrating String ProblemUniqueness of Solutions of the Wave EquationAppendixesBibliographySome Fourier Series ExpansionsSolutions of Some Regular Sturm-Liouville ProblemsIndex
