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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.

Inhaltsverzeichnis

Frontmatter

Distributions and their Basic Applications

Frontmatter

Chapter 1. Basic Definitions and Operations

Abstract
The notion of a distribution (or a generalized function—the term often used in other languages) is a comparatively recent invention, although the concept is one of the most important in mathematical areas with physical applications. By the middle of the 20th century, the theory took final shape, and distributions are commonly used by physicists and engineers today.
Alexander I. Saichev, Wojbor A. Woyczyński

Chapter 2. Basic Applications: Rigorous and Pragmatic

Abstract
In this section we give a couple of seemingly naive physical examples. Keeping them in mind, however, reinforces appropriate intuitive images of distributions that help solidify a formal mathematical understanding of the theory, and make it easier to grasp the automation of computations that can be achieved with help of distribution theory.
Alexander I. Saichev, Wojbor A. Woyczyński

Integral Transforms and Divergent Series

Frontmatter

Chapter 3. Fourier Transform

Abstract
In this chapter we study the Fourier transform and investigate its properties for functions f(t) depending on a single variable t which will be interpreted as time. The Fourier transform (or Fourier image) \(\tilde f\left( \omega \right)\)of f(t) is defined by
$$\tilde f\left( \omega \right) = {1 \over {2\pi }}\int {f\left( t \right)\mathop e\nolimits^{ - iwt} dt,} $$
(1)
whenever the integral on the right-hand side exists.
Alexander I. Saichev, Wojbor A. Woyczyński

Chapter 4. Asymptotics of Fourier Transforms

Abstract
In Chapter 3 we demonstrated that the Fourier transform \(\tilde f\left( \omega \right)\) of a smooth function f(t) rapidly decays to zero as ω → ∞. However, smoothness is rare in natural phenomena and one often encounters processes that are either discontinuous or violate the smoothness assumption in other ways. Such phenomena include, for example, shock fronts generated by large amplitude acoustic waves, ocean waves, or desert dunes with their characteristic sharp crests. These and many other examples explain the importance of the Fourier analysis of nonsmooth processes.
Alexander I. Saichev, Wojbor A. Woyczyński

Chapter 5. Stationary Phase and Related Methods

Abstract
In this chapter we will use methods developed in Chapter 4 to provide a general scheme for finding asymptotics. The remarkable Kelvin’s method of stationary phase will be employed as well.
Alexander I. Saichev, Wojbor A. Woyczyński

Chapter 6. Singular Integrals and Fractal Calculus

Abstract
This chapter is devoted to integrals similar to the familiar divergent Cauchy integral
$$\int {{{\varphi (s)} \over {s - x}}ds.} $$
(1)
Such integrals are often encountered in physical applications. If the function φ(s) does not vanish at s = x then the integrand in (1) has a nonintegrable singularity at that point. In practice physicists, using their intuition as a guide, often assign certain finite values to these integrals anyway. Then, it is a mathematician’s job to justify rigorously these “renormalizations of infinities”, translate additional physical requirements into mathematical terms and point out how different assumptions lead to different values of integral (1). The situation is fairly typical in collaboration of physicists and mathematicians.
Alexander I. Saichev, Wojbor A. Woyczyński

Chapter 7. Uncertainty Principle and Wavelet Transforms

Abstract
The method of wavelet transforms, which provides a decomposition of functions in terms of a fixed family of functions of constant shape but varying scales and locations, recently acquired broad significance in the analysis of signals and of experimental data from various physical phenomena. It is clear that the potential of this method has not yet been fully tapped. Nevertheless, its value for the whole spectrum of problems in many areas of science and engineering, including the study of electromagnetic and turbulent hydrodynamic fields, image reconstruction algorithms, prediction of earthquakes and tsunami waves, and statistical analysis of economic data, is by now quite obvious.
Alexander I. Saichev, Wojbor A. Woyczyński

Chapter 8. Summation of Divergent Series and Integrals

Abstract
The theory of distributions has a flavor similar to the theory of summation of divergent series and integrals and, as we have seen in Chapter 6, is closely related to the theory of singular integrals. A generic alternating series
$$1 - 1 + 1 - 1 + \ldots = \sum { \pm 1} $$
is a good example here. It seems to make no sense to assign a specific value to this infinite sum. Nevertheless, mathematicians have produced certain reasonable rules of summation that assign to it value 1/2. Such an assignment is in complete agreement with the intuition of physicists who encounter similar series. In this chapter, we will see how one can sum this, or even more strange, divergent series and integrals. To gain a better insight into the essence of this problem, let us begin with elementary examples and recall basic notions and theorems of the ordinary theory of convergent infinite series.
Alexander I. Saichev, Wojbor A. Woyczyński

Backmatter

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