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1 Overture 1.1 Some equations of mathematical physics 1.2 Linear differential operators 1.3 Separation of variables 2 Fourier Series 2.1 The Fourier series of a periodic function 2.2 A convergence theorem 2.3 Derivatives, integrals, and uniform convergence 2.4 Fourier series on intervals 2.5 Some applications 2.6 Further remarks on Fourier series 3 0rthogonal Sets of Functions 3.1 Vectors and inner products 3.2 Functions and inner products 3.3 Convergence and completeness 3.4 More about L2 spaces; the dominated convergence theorem 3.5 Regular Sturm-LiouviUe problems 3.6 Singular Sturm-Liouville problems 4 Some Boundary Value Problems 4.1 Some useful techniques 4.2 One-dimensional heat flow 4.3 One-dimensional wave motion 4.4 The Dirichlct problem 4.5 Multiple Fourier series and applications 5 Bessel Functions 5.1 Solutions of Bessel's equation 5.2 Bessel function identities 5.3 Asymptotics and zeros of Bessel functions 5.40 rthogonal sets of Bessel functions 5.5 Applications of Bessel functions 5.6 Variants of Bessel functions 6 0rthogonal Polynomials 6.1 Introduction 6.2 Legendrepolynomials 6.3 Spherical coordinates and Legendre functions 6.4 Hermite polynomials 6.5 Laguerre polynomials 6.6 Other orthogonal bases 7 The Fourier Transform 7.1 Convolutions 7.2 The Fourier transform 7.3 Some applications 7.4 Fourier transforms and Sturm-LiouviUe problems 7.5 Multivariable convolutions and Fourier transforms 7.6 Transforms related to the Fourier transform 8 The Laplace Transform 8.1 The Laplace transform 8.2 The inversion formula 8.3 Applications: Ordinary differential equations 8.4 Applications: Partial differential equations