| Introduction Notation Part I DISTRIBUTIONS AND THEIR BASIC APPLICATIONS 1 Basic Definitions and Operations 1.1 The "delta function" as viewed by a physicist and an engineer . 1.2 A rigorous definition of distributions 1.3 Singular distributions as limits of regular functions 1.4 Derivatives; linear operations 1.5 Multiplication by a smooth function; Leibniz formula 1.6 Integrals of distributions; the Heaviside function 1.7 Distributions of composite arguments 1.8 Convolution 1.9 The Dirac delta on Rn, lines and surfaces 1.10 Linear topological space of distributions 1.11 Exercises 2 Basic Applications: Rigorous and Pragmatic 2.1 Two generic physical examples 2.2 Systems governed by ordinary differential equations 2.3 One-dimensional waves 2.4 Continuity equation 2.5 Green's function of the continuity equation and Lagrangian coordinates 2.6 Method of characteristics 2.7 Density and concentration of the passive tracer 2.8 Incompressible medium 2.9 Pragmatic applications: beyond the rigorous theory of distributions 2.10 Exercises Part lI INTEGRAL TRANSFORMS AND DIVERGENT SERIE~ 3 Fourier Transform 3.1 Definition and elementary properties 3.2 Smoothness, inverse transform and convolution 3.3 Generalized Fourier transform 3.4 Transport equation 3.5 Exercises 4 Asymptotics of Fourier Transforms 4.1 Asymptotic notation, or how to get a camel to pass through a needle's eye 4.2 Riemann-Lebesgue Lemma 4.3 Functions with jumps 4.4 Gamma function and Fourier transforms of power functions . 4.5 Generalized Fourier transforms of power functions 4.6 Discontinuities of the second kind 4.7 Exercises 5 Stationary Phase and Related Method 5.1 Finding asymptotics: a general scheme 5.2 Stationary phase method 5.3 Fresnel approximation 5.4 Accuracy of the stationary phase method 5.5 Method of steepest descent 5.6 Exercises 6 Singular Integrals and Fractal Calculus 6.1 Principal value distribution 6.2 Principal value of Cauchy integral 6.3 A study of monochromatic wave 6.4 The Cauchy formula 6.5 The Hilbert transform 6.6 Analytic signals 6.7 Fourier transform of Heaviside function 6.8 Fractal integration 6.9 Fractal differentiation 6.10 Fractal relaxation 6.11 Exercises 7 Uncertainty Principle and Wavelet Transforms 8 Summation of Divergent Series and Integrals A Answers and Solutions B Bibliographical Notes Index |
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