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Contents

• V Full version
• 0 Introduction
• 1 Vector spaces
• 2 Fourier series
• 2.1 Fourier series
• 2.2 Convergence of Fourier series
• 2.3 Differentiability and Fourier series
• 3 Sturm-Liouville Theory
• 3.1 Sturm-Liouville operators
• 3.2 Least squares approximation
• 4 Partial differential equations
• 4.1 Laplace's equation
• 4.2 Laplace's equation in the unit disk in R2
• 4.3 Separation of variables
• 4.4 Laplace's equation in spherical polar coordinates
• 4.5 Multipole expansions for Laplace's equation
• 4.6 Laplace's equation in cylindrical coordinates
• 4.7 The heat equation
• 4.8 The wave equation
• 5 Distributions
• 5.1 Distributions
• 5.2 Green's functions
• 5.3 Green's functions for initial value problems
• 6 Fourier transforms
• 6.1 The Fourier transform
• 6.2 The Fourier inversion theorem
• 6.3 Parseval's theorem for Fourier transforms
• 6.4 A word of warning
• 6.5 Fourier transformation of distributions
• 6.6 Linear systems and response functions
• 6.7 General form of transfer function
• 6.8 The discrete Fourier transform
• 6.9 The fast Fourier transform*
• 7 More partial differential equations
• 7.1 Well-posedness
• 7.2 Method of characteristics
• 7.3 Characteristics for 2nd order partial differential equations
• 7.4 Green's functions for PDEs on Rn
• 7.5 Poisson's equation