# Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms and Wavelets: 1

## Saichev, Alexander I.; Woyczynski, Wojbor A.

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A comprehensive exposition on analytic methods for solving science and engineering problems, written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important to practioners and researchers. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise.

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Contenuti:

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.

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## 1.DISTRIBUTIONS IN THE PHYSICAL AND ENGINEERING SCIENCES , VOL - 1

ISBN 10: 0817639241 ISBN 13: 9780817639242
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Descrizione libro Condizione libro: Brand New. Brand New Original US Edition, Perfect Condition. Printed in English. Excellent Quality, Service and customer satisfaction guaranteed!. Codice libro della libreria AIND-16801

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## 2.DISTRIBUTIONS IN THE PHYSICAL AND ENGINEERING SCIENCES , VOL - 1

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Descrizione libro Condizione libro: New. Brand New Original US Edition.We Ship to PO BOX Address also. EXPEDITED shipping option also available for faster delivery. Codice libro della libreria AUSBNEW-16801

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## 3.Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms and Wavelets (Applied and Numerical Harmonic Analysis)

Editore: Birkhäuser (1996)
ISBN 10: 0817639241 ISBN 13: 9780817639242
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Descrizione libro Birkhäuser, 1996. Hardcover. Condizione libro: New. 1996. Codice libro della libreria DADAX0817639241

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## 4.Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms and Wavelets (Applied and Numerical Harmonic Analysis)

Editore: Birkhäuser (1996)
ISBN 10: 0817639241 ISBN 13: 9780817639242
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Descrizione libro Birkh&auml;user, 1996. Hardcover. Condizione libro: New. book. Codice libro della libreria 0817639241

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## 5.Distributions In The Physical And Engineering Sciences , Vol - 1

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## 6.Distributions in the Physical and Engineering Sciences, Vol. 1

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## 7.Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integra

Editore: Birkhäuser (2017)
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Descrizione libro Birkhäuser, 2017. Hardcover. Condizione libro: New. This item is printed on demand. Codice libro della libreria P110817639241

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## 8.Distributions in the Physical and Engineering Sciences. Distributional and Fractal Calculus, Integral Transforms and Wavelets

Editore: BirkhÇÏuser (1996)
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Descrizione libro BirkhÇÏuser, 1996. Hardback. Condizione libro: NEW. 9780817639242 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Codice libro della libreria HTANDREE0285564

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## 9.Distributions in the Physical and Engineering Sciences, Volume 1: Distributional and Fractal Calculus, Integral Transforms and Wavelets

Editore: Birkhauser, Boston, MA (1997)
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Descrizione libro Birkhauser, Boston, MA, 1997. Hard Cover. Condizione libro: New. 6 1/2" x 9 1/2". New condition. 336 pages. Illustrated. Codice libro della libreria S49-009117

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## 10.Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms and Wavelets

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