Front Cover -- Fourier Analysis and Boundary Value Problems -- Copyright Page -- Table of Contents -- Preface -- CHAPTER 1. A HEATED DISCUSSION -- 1.1 Historical Prologue -- 1.2 The Heat Equation -- 1.3 Boundary Value Problems -- 1.4 The Method of Separation of Variables -- 1.5 Linearity and Superposition of Solutions -- 1.6 Historical Epilogue -- Exercises -- CHAPTER 2. FOURIER SERIES -- 2.1 Introduction -- 2.2 Fourier Series -- 2.3 The Riemann-Lebesgue Theorem -- 2.4 The Convergence of Fourier Series -- 2.5 Fourier Series on Arbitrary Intervals -- 2.6 The Gibbs Phenomenon -- 2.7 Fejér Sums -- 2.8 Integration of Fourier Series -- 2.9 Historical Epilogue -- Exercises -- CHAPTER 3. RETURN TO THE HEATED BAR -- 3.1 Existence of a Solution -- 3.2 Uniqueness and Stability of the Solution -- 3.3 Nonzero Temperature at the Endpoints -- 3.4 Bar Insulated at the Endpoints -- 3.5 Mixed Endpoint Conditions -- 3.6 Heat Convection at One Endpoint -- 3.7 Time-Independent Problems -- 3.8 The Steady-State Solution -- 3.9 The Transient Solution -- 3.10 The Complete Solution -- 3.11 Time-Dependent Problems -- Exercises -- CHAPTER 4. GENERALIZED FOURIER SERIES -- 4.1 Sturm-Liouville Problems -- 4.2 The Eigenvalues and Eigenfunctions -- 4.3 The Existence of the Eigenvalues -- 4.4 Generalized Fourier Series -- 4.5 Approximations -- 4.6 Historical Epilogue -- Exercises -- CHAPTER 5. THE WAVE EQUATION -- 5.1 Introduction -- 5.2 The Vibrating String -- 5.3 D'Alembert's Solution -- 5.4 A Struck String -- 5.5 Bernoulli's Solution -- 5.6 Time-Independent Problems -- 5.7 Time-Dependent Problems -- 5.8 Historical Epilogue -- Exercises -- CHAPTER 6. ORTHOGONAL SYSTEMS -- 6.1 Fourier Series and Parseval's Identity -- 6.2 An Approximation Problem -- 6.3 The Uniform Convergence of Fourier Series -- 6.4 Convergence in the Mean -- 6.5 Applications to the Vibrating String.

6.6 The Riesz-Fischer Theorem -- Exercises -- CHAPTER 7. FOURIER TRANSFORMS -- 7.1 The Laplace Equation -- 7.2 Fourier Transforms -- 7.3 Properties of the Fourier Transform -- 7.4 Convolution -- 7.5 Solution of the Dirichlet Problem for the Half-Plane -- 7.6 The Fourier Transform Method -- Exercises -- CHAPTER 8. LAPLACE TRANSFORMS -- 8.1 The Laplace Transform and the Inversion Theorem -- 8.2 Properties of the Laplace Transform -- 8.3 Convolution -- 8.4 The Telegraph Equation -- 8.5 The Method of Residues -- 8.6 Historical Epilogue -- Exercises -- CHAPTER 9. BOUNDARY VALUE PROBLEMS IN HIGHER DIMENSIONS -- 9.1 Electrostatic Potential in a Charged Box -- 9.2 Double Fourier Series -- 9.3 The Dirichlet Problem in a Box -- 9.4 Return to the Charged Box -- 9.5 The Multiple Fourier Transform Method -- 9.6 The Double Laplace Transform Method -- Exercises -- CHAPTER 10. BOUNDARY VALUE PROBLEMS WITH CIRCULAR SYMMETRY -- 10.1 Vibrations of a Circular Membrane -- 10.2 The Gamma Function -- 10.3 Bessel Functions of the First Kind -- 10.4 Recursion Formulas for Bessel Functions -- 10.5 Bessel Functions of the Second Kind -- 10.6 The Zeros of Bessel Functions -- 10.7 Orthogonal Systems of Bessel Functions -- 10.8 Fourier-Bessel Series and Dini-Bessel Series -- 10.9 Return to the Vibrating Membrane -- 10.10 Modified Bessel Functions -- 10.11 The Skin Effect -- Exercises -- CHAPTER 11. BOUNDARY VALUE PROBLEMS WITH SPHERICAL SYMMETRY -- 11.1 The Potbellied Stove -- 11.2 Solutions of the Legendre Equation -- 11.3 The Norms of the Legendre Polynomials -- 11.4 Fourier-Legendre Series -- 11.5 Return to the Potbellied Stove -- 11.6 The Dirichlet Problem for the Sphere -- 11.7 The Associated Legendre Functions -- 11.8 Solution of the Dirichlet Problem for the Sphere -- 11.9 Poisson's Integral Formula for the Sphere -- 11.10 The Cooling of a Sphere -- Exercises.

CHAPTER 12. DISTRIBUTIONS AND GREEN'S FUNCTIONS -- 12.1 Historical Prologue -- 12.2 Distributions -- 12.3 Basic Properties of Distributions -- 12.4 Differentiation of Distributions -- 12.5 Sequences and Series of Distributions -- 12.6 Convolution -- 12.7 The Poisson Equation on the Sphere -- 12.8 Distributions Depending on a Parameter -- 12.9 The Cauchy Problem for Time-Dependent Equations -- 12.10 Conclusion -- Exercises -- APPENDIX A. UNIFORM CONVERGENCE -- Excercise -- APPENDIX B. IMPROPER INTEGRALS -- Exercises -- APPENDIX C. TABLES OF FOURIER AND LAPLACE TRANSFORMS -- APPENDIX D. HISTORICAL BIBLIOGRAPHY -- Index.