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This exceptionally well-written and well-organized text is the outgrowth of a course given every year for 45 years at the Chalmers University of Technology, Goteborg, Sweden. The object of the course was to give students a basic knowledge of Fourier analysis and certain of its applications. The text is self-contained with respect to such analysis; however, in areas where the author relies on results from branches of mathematics outside the scope of this book, references to widely used books are given.
nTable of Contents:
nChapter 1. Fourier series
n1.1 Basic concepts
n1.2 Fourier series and Fourier coefficients
n1.3 A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem
n1.4 Convergence of Fourier series
n1.5 The Parseval formula
n1.6 Determination of the sum of certain trigonometric series
nChapter 2. Orthogonal systems
n2.1 Integration of complex-valued functions of a real variable
n2.2 Orthogonal systems
n2.3 Complete orthogonal systems
n2.4 Integration of Fourier series
n2.5 The Gram-Schmidt orthogonalization process
n2.6 Sturm-Liouville problems
nChapter 3. Orthogonal polynomials
n3.1 The Legendre polynomials
n3.2 Legendre series
n3.3 The Legendre differential equation. The generating function of the Legendre polynomials
n3.4 The Tchebycheff polynomials
n3.5 Tchebycheff series
n3.6 The Hermite polynomials. The Laguerre polynomials
nChapter 4. Fourier transforms
n4.1 Infinite interval of integration
n4.2 The Fourier integral formula: a heuristic introduction
n4.3 Auxiliary theorems
n4.4 Proof of the Fourier integral formula. Fourier transforms
n4.5 The convention theorem. The Parseval formula
nChapter 5. Laplace transforms
n5.1 Definition of the Laplace transform. Domain. Analyticity
n5.2 Inversion formula
n5.3 Further properties of Laplace transforms. The convolution theorem
n5.4 Applications to ordinary differential equations
nChapter 6. Bessel functions
n6.1 The gamma function
n6.2 The Bessel differential equation. Bessel functions
n6.3 Some particular Bessel functions
n6.4 Recursion formulas for the Bessel functions
n6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions
n6.6 Bessel series
n6.7 The generating function of the Bessel functions of integral order
n6.8 Neumann functions
nChapter 7. Partial differential equations of first order
n7.1 Introduction
n7.2 The differential equation of a family of surfaces
n7.3 Homogeneous differential equations
n7.4 Linear and quasilinear differential equations
nChapter 8. Partial differential equations of second order
n8.1 Problems in physics leading to partial differential equations
n8.2 Definitions
n8.3 The wave equation
n8.4 The heat equation
n8.5 The Laplace equation
nAnswers to exercises; Bibliography; Conventions; Symbols; Index
nWritten on an advanced level, the book is aimed at advanced undergraduates and graduate students with a background in calculus, linear algebra, ordinary differential equations, and complex analysis. Over 260 carefully chosen exercises, with answers, encompass both routing and more challenging problems to help students test their grasp of the material.

Introduction to Partial Differential Equations

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