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| Preface | |
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| Fourier Series | |
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| Piecewise Continuous Functions | |
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| Fourier Cosine Series | |
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| Examples | |
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| Fourier Sine Series | |
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| Examples | |
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| Fourier Series | |
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| Examples | |
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| Adaptations to Other Intervals | |
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| Convergence of Fourier Series | |
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| One-Sided Derivatives | |
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| A Property of Fourier Coefficients | |
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| Two Lemmas | |
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| A Fourier Theorem | |
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| A Related Fourier Theorem | |
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| Examples | |
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| Convergence on Other Intervals | |
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| A Lemma | |
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| Absolute and Uniform Convergence of Fourier Series | |
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| The Gibbs Phenomenon | |
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| Differentiation of Fourier Series | |
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| Integration of Fourier Series | |
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| Partial Differential Equations of Physics | |
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| Linear Boundary Value Problems | |
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| One-Dimensional Heat Equation | |
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| Related Equations | |
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| Laplacian in Cylindrical and Spherical Coordinates | |
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| Derivations | |
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| Boundary Conditions | |
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| Duhamel's Principle | |
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| A Vibrating String | |
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| Vibrations of Bars and Membranes | |
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| General Solution of the Wave Equation | |
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| Types of Equations and Boundary Conditions | |
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| The Fourier Method | |
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| Linear Operators | |
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| Principle of Superposition | |
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| Examples | |
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| Eigenvalues and Eigenfunctions | |
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| A Temperature Problem | |
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| A Vibrating String Problem | |
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| Historical Development | |
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| Boundary Value Problems | |
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| A Slab with Faces at Prescribed Temperatures | |
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| Related Temperature Problems | |
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| Temperatures in a Sphere | |
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| A Slab with Internally Generated Heat | |
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| Steady Temperatures in Rectangular Coordinates | |
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| Steady Temperatures in Cylindrical Coordinates | |
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| A String with Prescribed Initial Conditions | |
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| Resonance | |
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| An Elastic Bar | |
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| Double Fourier Series | |
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| Periodic Boundary Conditions | |
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| Fourier Integrals and Applications | |
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| The Fourier Integral Formula | |
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| Dirichlet's Integral | |
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| Two Lemmas | |
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| A Fourier Integral Theorem | |
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| The Cosine and Sine Integrals | |
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| Some Eigenvalue Problems on Undounded Intervals | |
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| More on Superposition of Solutions | |
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| Steady Temperatures in a Semi-Infinite Strip | |
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| Temperatures in a Semi-Infinite Solid | |
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| Temperatures in an Unlimited Medium | |
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| Orthonormal Sets | |
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| Inner Products and Orthonormal Sets | |
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| Examples | |
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| Generalized Fourier Series | |
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| Examples | |
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| Best Approximation in the Mean | |
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| Bessel's Inequality and Parseval's Equation | |
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| Applications to Fourier Series | |
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| Sturm-Liouville Problems and Applications | |
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| Regular Sturm-Liouville Problems | |
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| Modifications | |
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| Orthogonality of Eigenfunctions adn Real Eigenvalues | |
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| Real-Valued Eigenfunctions | |
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| Nonnegative Eigenvalues | |
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| Methods of Solution | |
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| Examples of Eigenfunction Expansions | |
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| A Temperature Problem in Rectangular Coordinates | |
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| Steady Temperatures | |
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| Other Coordinates | |
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| A Modification of the Method | |
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| Another Modification | |
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| A Vertically Hung Elastic Bar | |
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| Bessel Functions and Applications | |
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| The Gamma Function | |
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| Bessel Functions Jn(x) | |
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| Solutions When v = 0,1,2,. | |
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| Recurrence Relations | |
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| Bessel's Integral Form | |
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| Some Consequences of the Integral Forms | |
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| The Zeros of Jn(x) | |
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| Zeros of Related Functions | |
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| Orthogonal Sets of Bessel Functions | |
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| Proof of the Theorems | |
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| Two Lemmas | |
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| Fourier-Bessel Series | |
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| Examples | |
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| Temperatures in a Long Cylinder | |
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| A Temperature Problem in Shrunken Fittings | |
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| Internally Generated Heat | |
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| Temperatures in a Long Cylindrical Wedge | |
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| Vibration of a Circular Membrane | |
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| Legendre Polynomials and Applications | |
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| Solutions of Legendre's Equation | |
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| Legendre Polynomials | |
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| Rodrigues' Formula | |
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| Laplace's Integral Form | |
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| Some Consequences of the Integral Form | |
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| Orthogonality of Legendre Polynomials | |
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| Normalized Legendre Polynomials | |
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| Legendre Series | |
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| The Eigenfunctions Pn(cos ) | |
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| Dirichlet Problems in Spherical Regions | |
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| Steady Temperatures in a Hemisphere | |
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| Verification of Solutions and Uniqueness | |
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| Abel's Test for Uniform Convergence | |
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| Verification of Solution of Temperature Problem | |
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| Uniqueness of Solutions of the Heat Equation | |
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| Verification of Solution of Vibrating String Problem | |
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| Uniqueness of Solutions of the Wave Equation | |
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| Appendixes | |
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| Bibliography | |
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| Some Fourier Series Expansions | |
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| Solutions of Some Regular Sturm-Liouville Problems | |
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| Some Fourier-Bessel Series Expansions | |
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| Index | |