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Distributions in the Physical and Engineering Sciences is a comprehensive exposition on analytic methods for solving science and engineering problems which is written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important to practioners and researchers. The goal of the book is to give the reader, specialist and non-specialist useable and modern mathematical tools in their research and analysis. This new text is intended for graduate students and researchers in applied mathematics, physical sciences and engineering. The careful explanations, accessible writing style, and many illustrations/examples also make it suitable for use as a self-study reference by anyone seeking greater understanding and proficiency in the problem solving methods presented. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise.
Content Level »Research
Keywords »Fourier transform - Haar wavelet - Mathematics - Physics - Signal - Singular integral - Wavelet - calculus - integral transform
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.