Distributions in the Physical and Engineering Sciences

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