Fourier Series and Boundary Value Problems
by Brown, James Ward; Churchill, Ruel VanceEdition: 6th
Format: Hardcover
Pub. Date: 2000-08-02
Publisher(s): McGraw-Hill Science/Engineering/Math
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Summary
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.
Author Biography
James Ward Brown is Professor of Mathematics at The University of Michigan--Dearborn Ruel V. Churchill was, at the time of his death in 1987, Professor Emeritus of Mathematics at The University of Michigan, where he began teaching in 1922
Table of Contents
Preface1 Partial Differential Equations of PhysicsLinear Boundary Value ProblemsConduction of HeatHigher DimensionsCylindrical CoordinatesSpherical CoordinatesBoundary ConditionsA Vibrating StringVibrations of Bars and MembranesTypes of Equations and Boundary ConditionsMethods of Solution2 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Linear Boundary Value ProblemsConduction of HeatHigher DimensionsCylindrical CoordinatesSpherical CoordinatesBoundary ConditionsA Vibrating StringVibrations of Bars and MembranesTypes of Equations and Boundary ConditionsMethods of Solution2 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Higher DimensionsCylindrical CoordinatesSpherical CoordinatesBoundary ConditionsA Vibrating StringVibrations of Bars and MembranesTypes of Equations and Boundary ConditionsMethods of Solution2 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Spherical CoordinatesBoundary ConditionsA Vibrating StringVibrations of Bars and MembranesTypes of Equations and Boundary ConditionsMethods of Solution2 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A Vibrating StringVibrations of Bars and MembranesTypes of Equations and Boundary ConditionsMethods of Solution2 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Types of Equations and Boundary ConditionsMethods of Solution2 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
2 The Fourier MethodLinear OperatorsPrinciple of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Principle of SuperpositionA GeneralizationA Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A Temperature ProblemThe Nonhomogeneous ConditionA Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A Vibrating String ProblemThe Nonhomogeneous ConditionHistorical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Historical Development3 Orthonormal Sets and Fourier SeriesPiecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Piecewise Continuous FunctionsInner Products and Orthonormal SetsExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
ExamplesGeneralized Fourier SeriesFourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Fourier Cosine SeriesFourier Sine SeriesFourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Fourier SeriesExamplesBest Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Best Approximation in the MeanBessel's Inequality and a Property of Fourier Constants4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
4 Convergence of Fourier SeriesOne-Sided DerivativesTwo LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Two LemmasA Fourier TheoremDiscussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Discussion of the Theorem and Its CorollaryFourier Series on Other IntervalsA LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A LemmaUniform Convergence of Fourier SeriesDifferentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Differentiation of Fourier SeriesIntegration of Fourier SeriesConvergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Convergence in the Mean5 Boundary Value ProblemsA Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A Slab with Faces at Prescribed TemperaturesRelated ProblemsA Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A Slab with Internally Generated HeatA Dirichlet ProblemCylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Cylindrical CoordinatesA String with Prescribed Initial VelocityResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
ResonanceAn Elastic BarDouble Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Double Fourier SeriesPeriodic Boundary Conditions6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
6 Sturm-Liouville Problems and ApplicationsRegular Sturm-Liouville ProblemsModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
ModificationsOrthogonality of EigenfunctionsReal-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Real-Valued Eigenfunctions and Nonnegative EigenvaluesMethods of SolutionExamples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Examples of Eigenfunction ExpansionsSurface Heat TransferA Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A Dirichlet ProblemModifications of the MethodA Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
A Vertically Hung Elastic Bar7 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
The 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 Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Two LemmasA Fourier Integral TheoremThe Cosine and Sine IntegralsMore on Superposition of SolutionsTemperatures in a Semi-Infinite SolidTemperatures in an Unlimited Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
The Cosine and Sine IntegralsMore on Superposition of SolutionsTemperatures in a Semi-Infinite SolidTemperatures in an Unlimited Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Temperatures in a Semi-Infinite SolidTemperatures in an Unlimited Medium8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
8 Bessel Functions and ApplicationsBessel Functions JnGeneral Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
General Solutions of Bessel's EquationRecurrence RelationsBessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Bessel's Integral FormThe Zeros of J0(x)Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Zeros of Related FunctionsOrthogonal Sets of Bessel FunctionsProof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Proof of TheoremThe Orthonormal FunctionsFourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Fourier-Bessel SeriesTemperatures in a Long CylinderInternally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Internally Generated HeatVibration of a Circular Membrane9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
9 Legendre Polynomials and ApplicationsSolutions of Legendre's EquationLegendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Legendre PolynomialsOrthogonality of Legendre PolynomialsRodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Rodrigues' Formula and NormsLegendre SeriesThe Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
The Eigenfunctions Pn(cos theta)Dirichlet Problems in Spherical RegionsSteady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Steady Temperatures in a Hemisphere10 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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Abel'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 EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Uniqueness of Solutions of the Heat EquationVerification of Solution of Vibrating String ProblemUniqueness of Solutions of the Wave EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
Uniqueness of Solutions of the Wave EquationOn Laplace's and Poisson's EquationsAn ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
An ExampleAppendixes1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
1 Bibliography2 Some Fourier Series Expansions3 Solutions of Some Regular Sturm-Liouville Problems
3 Solutions of Some Regular Sturm-Liouville Problems
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