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Chapter 4
Fourier series and PDEs

 4.1 Boundary value problems
  4.1.1 Boundary value problems
  4.1.2 Eigenvalue problems
  4.1.3 Orthogonality of eigenfunctions
  4.1.4 Fredholm alternative
  4.1.5 Application
  4.1.6 Exercises
 4.2 The trigonometric series
  4.2.1 Periodic functions and motivation
  4.2.2 Inner product and eigenvector decomposition
  4.2.3 The trigonometric series
  4.2.4 Exercises
 4.3 More on the Fourier series
  4.3.1 2L -periodic functions
  4.3.2 Convergence
  4.3.3 Differentiation and integration of Fourier series
  4.3.4 Rates of convergence and smoothness
  4.3.5 Exercises
 4.4 Sine and cosine series
  4.4.1 Odd and even periodic functions
  4.4.2 Sine and cosine series
  4.4.3 Application
  4.4.4 Exercises
 4.5 Applications of Fourier series
  4.5.1 Periodically forced oscillation
  4.5.2 Resonance
  4.5.3 Exercises
 4.6 PDEs, separation of variables, and the heat equation
  4.6.1 Heat on an insulated wire
  4.6.2 Separation of variables
  4.6.3 Insulated ends
  4.6.4 Exercises
 4.7 One dimensional wave equation
  4.7.1 Exercises
 4.8 D’Alembert solution of the wave equation
  4.8.1 Change of variables
  4.8.2 D’Alembert’s formula
  4.8.3 Another way to solve for the side conditions
  4.8.4 Exercises
 4.9 Steady state temperature and the Laplacian
  4.9.1 Exercises
 4.10 Dirichlet problem in the circle and the Poisson kernel
  4.10.1 Laplace in polar coordinates
  4.10.2 Series solution
  4.10.3 Poisson kernel
  4.10.4 Exercises