Fourier analysis and its applications

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Wadsworth & Brooks/Cole Advanced Books & Software, 1992 - Mathematics - 433 pages
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1. OVERTURE Some Equations of Mathematical Physics / Linear Differential Operators / Separation of Variables 2. FOURIER SERIES The Fourier Series of a Periodic Function / A Convergence Theorem / Derivatives, Integrals, and Uniform Convergence / Fourier Series on Intervals / Some Applications / Further Remarks on Fourier Series 3. ORTHOGONAL SETS OF FUNCTIONS Vectors and Inner Products / Functions and Inner Productions / Convergence and Completeness / More about L[2] Spaces; The Dominated Convergence Theorem / Regular Sturm-Liouville Problems / Singular Sturm-Liouville Problems 4. SOME BOUNDARY VALUE PROBLEMS Some Useful Techniques / One-Dimensional Heat Flow / One-Dimensional Wave Motion / The Dirichlet Problem / Multiple Fourier Series and Applications 5. BESSEL FUNCTIONS Solution of Bessel's Equation / Bessel Function Identities / Asymptotics and Zeros of Bessel Functions / Orthogonal Sets of Bessel Functions / Applications of Bessel Functions / Variants of Bessel Functions 6. ORTHOGONAL POLYNOMIALS Introduction / Legendre Polynomials / Spherical Coordinates and Legendre Functions / Hermite Polynomials / Laguerre Polynomials / Other Orthogonal Bases 7. THE FOURIER TRANSFORM Convolutions / The Fourier Transform / Some Applications / Fourier Transforms and Sturm-Liouville Multivariable Convolutions and Fourier Transforms / Transforms Related to the Fourier Transform 8. THE LAPLACE TRANSFORM The Laplace Transform / The Inversion Formula / Applications: Ordinary Differential Equations / Applications: Partial Differential Equations / Applications: Integral Equations / Asymptotics of Laplace Transforms 9. GENERALIZED FUNCTIONS Distribution / Convergence, Convolution, and Approximation / More Examples: Periodic Distributions and Finite Parts / Tempered Distributions and Fourier Transforms / Weak Solutions of Differential Equations 10. GREEN'S FUNCTIONS Green's Functions for Ordinary Differential Operators / Green's Functions for Partial Differential Operators / Green's Functions and Regular Sturm-Liouville Problems / Green's Functions and Singular Sturm-Liouville Problems / APPENDICES / 1. SOME PHYSICAL DERIVATIONS / 2. SUMMARY OF COMPLEX VARIABLE THEORY / 3. THE GAMMA FUNCTION / 4. CALCULATIONS IN POLAR COORDINATES / 5. THE FUNDAMENTAL THEOREM OF ORDINARY DIFFERENTIAL EQUATIONS / ANSWERS TO THE EXERCISES / REFERENCES / INDEX OF SYMBOLS / INDEX

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Contents

Overture
1
Fourier Series
18
Orthogonal Sets of Functions
62
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