Fourier Analysis and Its Applications / Edition 1

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This book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.

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Editorial Reviews

A textbook for an advanced undergraduate course for students of mathematics, physics, and engineering. Explains the cluster of techniques whose popularity and usefulness in science rivals that of sliced bread (with which it also shares some fundamental concepts). Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780534170943
  • Publisher: Cengage Learning
  • Publication date: 1/28/1992
  • Series: Advanced Mathematics Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 448
  • Product dimensions: 6.40 (w) x 9.30 (h) x 0.90 (d)

Table of Contents

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 L² 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, andApproximation / 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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