| Fourier Series | |
| Definition of Fourier series | p. 1 |
| Orthogonality of sines and cosines | p. 2 |
| Determination of the coefficients | p. 3 |
| Series of cosines and series of sines | p. 6 |
| Examples | p. 8 |
| Magnitude of coefficients under special hypotheses | p. 11 |
| Riemann's theorem on limit of general coefficient | p. 14 |
| Evaluation of a sum of cosines | p. 17 |
| Integral formula for partial sum of Fourier series | p. 17 |
| Convergence at a point of continuity | p. 18 |
| Uniform convergence under special hypotheses | p. 21 |
| Convergence at a point of discontinuity | p. 22 |
| Sufficiency of conditions relating to a restricted neighborhood | p. 24 |
| Weierstrass's theorem on trigonometric approximation | p. 25 |
| Least-square property | p. 27 |
| Parseval's theorem | p. 29 |
| Summation of series | p. 31 |
| Fejer's theorem for a continuous function | p. 32 |
| Proof of Weierstrass's theorem by means of de la Vallee Poussin's integral | p. 35 |
| The Lebesgue constants | p. 40 |
| Proof of uniform convergence by the method of Lebesgue | p. 42 |
| Legendre Polynomials | |
| Preliminary orientation | p. 45 |
| Definition of the Legendre polynomials by means of the generating function | p. 45 |
| Recurrence formula | p. 46 |
| Differential equation and related formulas | p. 48 |
| Orthogonality | p. 50 |
| Normalizing factor | p. 51 |
| Expansion of an arbitrary function in series | p. 53 |
| Christoffel's identity | p. 54 |
| Solution of the differential equation | p. 55 |
| Rodrigues's formula | p. 57 |
| Integral representation | p. 58 |
| Bounds of P[subscript n](x) | p. 61 |
| Convergence at a point of continuity interior to the interval | p. 63 |
| Convergence at a point of discontinuity interior to the interval | p. 65 |
| Bessel Functions | |
| Preliminary orientation | p. 69 |
| Definition of J[subscript 0](x) | p. 69 |
| Orthogonality | p. 71 |
| Integral representation of J[subscript 0](x) | p. 74 |
| Zeros of J[subscript 0](x) and related functions | p. 76 |
| Expansion of an arbitrary function in series | p. 79 |
| Definition of J[subscript n](x) | p. 80 |
| Orthogonality: developments in series | p. 82 |
| Integral representation of J[subscript n](x) | p. 84 |
| Recurrence formulas | p. 85 |
| Zeros | p. 87 |
| Asymptotic formula | p. 87 |
| Orthogonal functions arising from linear boundary value problems | p. 88 |
| Boundary Value Problems | |
| Fourier series: Laplace's equation in an infinite strip | p. 91 |
| Fourier series: Laplace's equation in a rectangle | p. 95 |
| Fourier series: vibrating string | p. 96 |
| Fourier series: damped vibrating string | p. 100 |
| Polar coordinates in the plane | p. 101 |
| Fourier series: Laplace's equation in a circle; Poisson's integral | p. 103 |
| Transformation of Laplace's equation in three dimensions | p. 105 |
| Legendre series: Leplace's equation in a sphere | p. 107 |
| Bessel series: Laplace's equation in a cylinder | p. 109 |
| Bessel series: circular drumhead | p. 112 |
| Double Series; Laplace Series | |
| Boundary value problem in a cube; double Fourier series | p. 115 |
| General spherical harmonics | p. 118 |
| Laplace series | p. 121 |
| Harmonic polynomials | p. 126 |
| Rotation of axes | p. 129 |
| Integral representation for group of terms in the Laplace series | p. 132 |
| Completeness of the Laplace series | p. 137 |
| Boundary value problem in a cylinder; series involving Bessel functions of positive order | p. 138 |
| The Pearson Frequency Functions | |
| The Pearson differential equation | p. 142 |
| Quadratic denominator, real roots | p. 142 |
| Quadratic denominator, complex roots | p. 145 |
| Linear or constant denominator | p. 146 |
| Finiteness of moments | p. 147 |
| Orthogonal Polynomials | |
| Weight function | p. 149 |
| Schmidt's process | p. 151 |
| Orthogonal polynomials corresponding to an arbitrary weight function | p. 153 |
| Development of an arbitrary function in series | p. 155 |
| Formula of recurrence | p. 156 |
| Christoffel-Darboux identity | p. 157 |
| Symmetry | p. 158 |
| Zeros | p. 159 |
| Least-square property | p. 160 |
| Differential equation | p. 161 |
| Jacobi Polynomials | |
| Derivative definition | p. 166 |
| Orthogonality | p. 167 |
| Leading coefficients | p. 169 |
| Normalizing factor; series of Jacobi polynomials | p. 171 |
| Recurrence formula | p. 172 |
| Differential equation | p. 173 |
| Hermite Polynomials | |
| Derivative definition | p. 176 |
| Orthogonality and normalizing factor | p. 177 |
| Hermite and Gram-Charlier series | p. 178 |
| Recurrence formulas; differential equation | p. 179 |
| Generating function | p. 181 |
| Wave equation of the linear oscillator | p. 181 |
| Laguerre Polynomials | |
| Derivative definition | p. 184 |
| Orthogonality; normalizing factor; Laguerre series | p. 184 |
| Differential equation and recurrence formulas | p. 186 |
| Generating function | p. 187 |
| Wave equation of the hydrogen atom | p. 188 |
| Convergence | |
| Scope of the discussion | p. 191 |
| Magnitude of the coefficients; first hypothesis | p. 192 |
| Convergence; first hypothesis | p. 194 |
| Magnitude of the coefficients; second hypothesis | p. 197 |
| Convergence; second hypothesis | p. 199 |
| Special Jacobi polynomials | p. 200 |
| Multiplication or division of the weight function by a polynomial | p. 201 |
| Korous's theorem on bounds of orthonormal polynomials | p. 205 |
| Exercises | p. 209 |
| Bibliography | p. 229 |
| Index | p. 231 |
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