| |
| |
| Preface | |
| |
| |
| |
| Preliminary results in the integral calculus: series and integrals | |
| |
| |
| |
| Preliminary Results on Series | |
| |
| |
| |
| Summable series | |
| |
| |
| |
| Semi-convergent series | |
| |
| |
| |
| Preliminary Results on Integration | |
| |
| |
| |
| The Lebesgue integral | |
| |
| |
| |
| Improper semi-convergent Lebesgue integrals | |
| |
| |
| |
| Functions Represented by Series and Integrals | |
| |
| |
| |
| Functions represented by series | |
| |
| |
| |
| Functions represented by integrals | |
| |
| |
| Exercises for Chapter I | |
| |
| |
| |
| Elementary theory of distributions | |
| |
| |
| |
| Definition of Distributions | |
| |
| |
| |
| The vector space D | |
| |
| |
| |
| Distributions | |
| |
| |
| |
| The support of a distribution | |
| |
| |
| |
| Differentiation of Distributions | |
| |
| |
| |
| Definition | |
| |
| |
| |
| Examples of derivatives in the one-dimensional case | |
| |
| |
| |
| Examples of derivatives in the case of several variables | |
| |
| |
| |
| Multiplication of Distributions | |
| |
| |
| |
| Topology in Distribution Space. Convergence of Distributions. Series of Distributions | |
| |
| |
| |
| Distributions with Bounded Supports | |
| |
| |
| Exercises for Chapter II | |
| |
| |
| |
| Convolution | |
| |
| |
| |
| Tensor Product of Distributions | |
| |
| |
| |
| Tensor product of two distributions | |
| |
| |
| |
| Tensor product of several distributions | |
| |
| |
| |
| Convolution | |
| |
| |
| |
| Convolution of two distributions | |
| |
| |
| |
| Definition of the convolution product of several distributions. Associativity of convolution | |
| |
| |
| |
| Convolution equations | |
| |
| |
| |
| Convolution in Physics | |
| |
| |
| Exercises for Chapter III | |
| |
| |
| |
| Fourier series | |
| |
| |
| |
| Fourier Series of Periodic Functions and Distributions | |
| |
| |
| |
| Fourier series expansion of a periodic function | |
| |
| |
| |
| Fourier series expansion of a periodic distribution | |
| |
| |
| |
| Convergence of Fourier Series in the Distribution Sense and in the Function Sense | |
| |
| |
| |
| Convergence of the Fourier series of a distribution | |
| |
| |
| |
| Convergence of the Fourier series of a function | |
| |
| |
| |
| Hilbert Bases of a Hilbert Space. Mean-Square Convergence of a Fourier Series | |
| |
| |
| |
| Definition of a Hilbert space | |
| |
| |
| |
| Hilbert basis | |
| |
| |
| |
| The space L[superscript 2](T) | |
| |
| |
| |
| The Convolution Algebra D'([gamma]) | |
| |
| |
| Exercises for Chapter IV | |
| |
| |
| |
| The fourier transform | |
| |
| |
| |
| Fourier Transforms of Functions of One Variable | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Fourier transform | |
| |
| |
| |
| Fundamental relations and inequalities | |
| |
| |
| |
| Spaces s of infinitely differentiable functions with all derivatives decreasing rapidly | |
| |
| |
| |
| Fourier Transforms of Distributions in One Variable | |
| |
| |
| |
| Definition | |
| |
| |
| |
| Tempered distributions: the space y' | |
| |
| |
| |
| Fourier transforms of tempered distributions | |
| |
| |
| |
| The Parseval-Plancherel equation. Fourier transforms in L[superscript 2] | |
| |
| |
| |
| The Poisson summation formula | |
| |
| |
| |
| The Fourier transform: multiplication and convolution | |
| |
| |
| |
| Other expressions for the Fourier transform | |
| |
| |
| |
| Fourier Transforms in Several Variables | |
| |
| |
| |
| A Physical Application of the Fourier Transform: Solution of the Heat Conduction Equation | |
| |
| |
| Exercises for Chapter V | |
| |
| |
| |
| The laplace transform | |
| |
| |
| |
| Laplace Transforms of Functions | |
| |
| |
| |
| Laplace Transforms of Distributions | |
| |
| |
| |
| Definition | |
| |
| |
| |
| Examples of Laplace transforms | |
| |
| |
| |
| The Laplace transforms and convolution | |
| |
| |
| |
| Fourier and Laplace transform. Inversion of the Laplace transform | |
| |
| |
| |
| Applications of the Laplace Transform. Operational Calculus | |
| |
| |
| Exercises for Chapter VI | |
| |
| |
| |
| The wave and heat conduction equations | |
| |
| |
| |
| Equation of Vibrating Strings | |
| |
| |
| |
| Physical problems associated with the equation of vibrating strings | |
| |
| |
| |
| Solution of the equation of vibrating strings by the method of travelling waves. Cauchy's problem | |
| |
| |
| |
| Solution of Cauchy's problem by Fourier analysis | |
| |
| |
| |
| Vibrating Membranes and Waves in Three Dimensions | |
| |
| |
| |
| The solution of the vibrating membrane equation and the wave equation in three dimensions by the method of travelling waves. Cauchy problems | |
| |
| |
| |
| Solution of the Cauchy problem for vibrating membranes by the method of harmonics | |
| |
| |
| |
| Particular cases of rectangular and circular membranes | |
| |
| |
| |
| The wave equation in R[superscript n] | |
| |
| |
| |
| The Heat Conduction Equation | |
| |
| |
| |
| Solution by the method of propagation. Cauchy's problem | |
| |
| |
| |
| The solution of Cauchy's problem by the method of harmonics | |
| |
| |
| Exercises for Chapter VII | |
| |
| |
| |
| The gamma function | |
| |
| |
| |
| The Function [Gamma] (z) | |
| |
| |
| |
| The Function B (p, q) | |
| |
| |
| |
| The Complementary Formula | |
| |
| |
| |
| Generalization of the Beta Function | |
| |
| |
| |
| Graphical Representation of the Function y = [Gamma](x) for Real x | |
| |
| |
| |
| Stirling's Formula | |
| |
| |
| |
| Application to the Expansion of 1/[Gamma] as an Infinite Product | |
| |
| |
| |
| The Function [psi](z) = [Gamma]'(z)/[Gamma](z) | |
| |
| |
| |
| Applications | |
| |
| |
| Exercises for Chapter VIII | |
| |
| |
| |
| Bessel functions | |
| |
| |
| |
| Definitions and Elementary Properties | |
| |
| |
| |
| Definitions of the Bessel, Neumann and Hankel functions | |
| |
| |
| |
| Integral representations of Bessel functions | |
| |
| |
| |
| Recurrence relations | |
| |
| |
| |
| Other properties of Bessel functions | |
| |
| |
| |
| Formulae | |
| |
| |
| Exercises for Chapter IX | |
| |
| |
| Index | |