CONTENTS ix
Chapter 5/Fourier Series
5.1* The Coefficients 104
5.2* Even, Odd, Periodic, and Complex Functions 113
5.3* Orthogonality and General Fourier Series 118
5.4* Completeness 124
5.5 Completeness and the Gibbs Phenomenon 136
5.6 Inhomogeneous Boundary Conditions 147
Chapter 6/Harmonic Functions
6.1* Laplace’s Equation 152
6.2* Rectangles and Cubes 161
6.3* Poisson’s Formula 165
6.4 Circles, Wedges, and Annuli 172
(The next four chapters may be studied in any order.)
Chapter 7/Green’s Identities and Green’s Functions
7.1 Green’s First Identity 178
7.2 Green’s Second Identity 185
7.3 Green’s Functions 188
7.4 Half-Space and Sphere 191
Chapter 8/Computation of Solutions
8.1 Opportunities and Dangers 199
8.2 Approximations of Diffusions 203
8.3 Approximations of Waves 211
8.4 Approximations of Laplace’s Equation 218
8.5 Finite Element Method 222
Chapter 9/Waves in Space
9.1 Energy and Causality 228
9.2 The Wave Equation in Space-Time 234
9.3 Rays, Singularities, and Sources 242
9.4 The Diffusion and Schr
¨
odinger Equations 248
9.5 The Hydrogen Atom 254
Chapter 10/Boundaries in the Plane and in Space
10.1 Fourier’s Method, Revisited 258
10.2 Vibrations of a Drumhead 264
10.3 Solid Vibrations in a Ball 270
10.4 Nodes 278
10.5 Bessel Functions 282
