Distributions in the Physical and Engineering Sciences, Volume 1: Distributional and Fractal Calculu

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.- II Integral Transforms and Divergent Series.- 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.- 7.1 Functional Hilbert spaces.- 7.2 Time-frequency localization and the uncertainty principle.- 7.3 Windowed Fourier transform.- 7.4 Continuous wavelet transforms.- 7.5 Haar wavelets and multiresolution analysis.- 7.6 Continuous Daubechies' wavelets.- 7.7 Wavelets and distributions.- 7.8 Exercises.- 8 Summation of Divergent Series and Integrals.- 8.1 Zeno's "paradox" and convergence of infinite series.- 8.2 Summation of divergent series.- 8.3 Tiring Achilles and the principle of infinitesimal relaxation.- 8.4 Achilles chasing the tortoise in presence of head winds.- 8.5 Separation of scales condition.- 8.6 Series of complex exponentials.- 8.7 Periodic Dirac deltas.- 8.8 Poisson summation formula.- 8.9 Summation of divergent geometric series.- 8.10 Shannon's sampling theorem.- 8.11 Divergent integrals.- 8.12 Exercises.- A Answers and Solutions.- A.1 Chapter 1. Definitions and operations.- A.2 Chapter 2. Basic applications.- A.3 Chapter 3. Fourier transform.- A.4 Chapter 4. Asymptotics of Fourier transforms.- A.5 Chapter 5. Stationary phase and related methods.- A.6 Chapter 6. Singular integrals and fractal calculus.- A.7 Chapter 7. Uncertainty principle and wavelet transform.- A. 8 Chapter 8. Summation of divergent series and integrals.- B Bibliographical Notes.