| Preface | p. 11 |
| Preliminary results in the integral calculus: series and integrals | |
| Preliminary Results on Series | p. 13 |
| Summable series | p. 13 |
| Semi-convergent series | p. 23 |
| Preliminary Results on Integration | p. 26 |
| The Lebesgue integral | p. 26 |
| Improper semi-convergent Lebesgue integrals | p. 43 |
| Functions Represented by Series and Integrals | p. 46 |
| Functions represented by series | p. 46 |
| Functions represented by integrals | p. 57 |
| Exercises for Chapter I | p. 65 |
| Elementary theory of distributions | |
| Definition of Distributions | p. 71 |
| The vector space D | p. 71 |
| Distributions | p. 73 |
| The support of a distribution | p. 79 |
| Differentiation of Distributions | p. 80 |
| Definition | p. 80 |
| Examples of derivatives in the one-dimensional case | p. 82 |
| Examples of derivatives in the case of several variables | p. 85 |
| Multiplication of Distributions | p. 91 |
| Topology in Distribution Space. Convergence of Distributions. Series of Distributions | p. 94 |
| Distributions with Bounded Supports | p. 98 |
| Exercises for Chapter II | p. 100 |
| Convolution | |
| Tensor Product of Distributions | p. 110 |
| Tensor product of two distributions | p. 110 |
| Tensor product of several distributions | p. 112 |
| Convolution | p. 112 |
| Convolution of two distributions | p. 112 |
| Definition of the convolution product of several distributions. Associativity of convolution | p. 121 |
| Convolution equations | p. 123 |
| Convolution in Physics | p. 134 |
| Exercises for Chapter III | p. 140 |
| Fourier series | |
| Fourier Series of Periodic Functions and Distributions | p. 145 |
| Fourier series expansion of a periodic function | p. 145 |
| Fourier series expansion of a periodic distribution | p. 149 |
| Convergence of Fourier Series in the Distribution Sense and in the Function Sense | p. 152 |
| Convergence of the Fourier series of a distribution | p. 152 |
| Convergence of the Fourier series of a function | p. 154 |
| Hilbert Bases of a Hilbert Space. Mean-Square Convergence of a Fourier Series | p. 158 |
| Definition of a Hilbert space | p. 158 |
| Hilbert basis | p. 159 |
| The space L[superscript 2](T) | p. 160 |
| The Convolution Algebra D'([gamma]) | p. 164 |
| Exercises for Chapter IV | p. 170 |
| The fourier transform | |
| Fourier Transforms of Functions of One Variable | p. 178 |
| Introduction | p. 178 |
| Fourier transform | p. 179 |
| Fundamental relations and inequalities | p. 180 |
| Spaces s of infinitely differentiable functions with all derivatives decreasing rapidly | p. 182 |
| Fourier Transforms of Distributions in One Variable | p. 187 |
| Definition | p. 187 |
| Tempered distributions: the space y' | p. 188 |
| Fourier transforms of tempered distributions | p. 189 |
| The Parseval-Plancherel equation. Fourier transforms in L[superscript 2] | p. 194 |
| The Poisson summation formula | p. 196 |
| The Fourier transform: multiplication and convolution | p. 197 |
| Other expressions for the Fourier transform | p. 199 |
| Fourier Transforms in Several Variables | p. 200 |
| A Physical Application of the Fourier Transform: Solution of the Heat Conduction Equation | p. 205 |
| Exercises for Chapter V | p. 208 |
| The laplace transform | |
| Laplace Transforms of Functions | p. 215 |
| Laplace Transforms of Distributions | p. 217 |
| Definition | p. 217 |
| Examples of Laplace transforms | p. 218 |
| The Laplace transforms and convolution | p. 222 |
| Fourier and Laplace transform. Inversion of the Laplace transform | p. 224 |
| Applications of the Laplace Transform. Operational Calculus | p. 230 |
| Exercises for Chapter VI | p. 235 |
| The wave and heat conduction equations | |
| Equation of Vibrating Strings | p. 242 |
| Physical problems associated with the equation of vibrating strings | p. 242 |
| Solution of the equation of vibrating strings by the method of travelling waves. Cauchy's problem | p. 251 |
| Solution of Cauchy's problem by Fourier analysis | p. 270 |
| Vibrating Membranes and Waves in Three Dimensions | p. 280 |
| The solution of the vibrating membrane equation and the wave equation in three dimensions by the method of travelling waves. Cauchy problems | p. 281 |
| Solution of the Cauchy problem for vibrating membranes by the method of harmonics | p. 291 |
| Particular cases of rectangular and circular membranes | p. 293 |
| The wave equation in R[superscript n] | p. 298 |
| The Heat Conduction Equation | p. 298 |
| Solution by the method of propagation. Cauchy's problem | p. 298 |
| The solution of Cauchy's problem by the method of harmonics | p. 301 |
| Exercises for Chapter VII | p. 303 |
| The gamma function | |
| The Function [Gamma] (z) | p. 311 |
| The Function B (p, q) | p. 313 |
| The Complementary Formula | p. 315 |
| Generalization of the Beta Function | p. 317 |
| Graphical Representation of the Function y = [Gamma](x) for Real x | p. 318 |
| Stirling's Formula | p. 320 |
| Application to the Expansion of 1/[Gamma] as an Infinite Product | p. 322 |
| The Function [psi](z) = [Gamma]'(z)/[Gamma](z) | p. 326 |
| Applications | p. 327 |
| Exercises for Chapter VIII | p. 330 |
| Bessel functions | |
| Definitions and Elementary Properties | p. 334 |
| Definitions of the Bessel, Neumann and Hankel functions | p. 334 |
| Integral representations of Bessel functions | p. 341 |
| Recurrence relations | p. 343 |
| Other properties of Bessel functions | p. 345 |
| Formulae | p. 350 |
| Exercises for Chapter IX | p. 353 |
| Index | p. 357 |
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