Modern Mathematics for the Engineer: Second Series

Edwin F. Beckenbach, Magnus R. Hestenes

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486316123
• Topic: Technologie et ingénierie
• Subtopic: Ingénierie civile

This volume and its predecessor were conceived to advance the level of mathematical sophistication in the engineering community. The books particularly focus on material relevant to solving the kinds of mathematical problems regularly confronted by engineers. Suitable as a text for advanced undergraduate and graduate courses as well as a reference for professionals, Volume Two's three-part treatment covers mathematical methods, statistical and scheduling studies, and physical phenomena.
Contributions include chapters on chance processes and fluctuations by William Feller, Monte Carlo calculations in problems of mathematical physics by Stanislaw M. Ulam, and circle, sphere, symmetrization, and some classical physical problems by George Pólya. Additional topics include integral transforms, information theory, the numerical solution of elliptic and parabolic partial differential equations, and other subjects involving the intersection of engineering and mathematics.

PART 1

Mathematical Methods

1

From Delta Functions to Distributions

ARTHUR ERDÉLYI

PROFESSOR OF MATHEMATICS
CALIFORNIA INSTITUTE OF TECHNOLOGY

1.1 Introduction

In mathematical physics, one often encounters “impulsive” forces acting for a short time only. A unit impulse would be described by a function p(t) that vanishes outside a short interval and is such that

It is convenient to idealize such forces as “instantaneous” and to attempt to describe them by a function δ(t) that vanishes except for a single value of t which we take to be t = 0, is undefined for t = 0, and for which

Such a function, one convinces oneself, should possess the “sifting property”

for every continuous function ϕ, and the corresponding property (obtained by integration by parts)

for every k times continuously differentiable function ϕ.

Unfortunately, it can be proved that no function, in the sense of the mathematical definition of this term, possesses the sifting property. Nevertheless, “impulse functions” postulated to have these or other similar properties are being used with great success in applied mathematics and mathematical physics.

The use of such improper functions can be defended as a kind of short-hand, or else as a heuristic means; it can also be justified by an appropriate mathematical theory. In Sec. 1.2 we shall indicate briefly some theories that can be employed to justify the use of the delta function. In order to provide a theoretical framework accommodating the great variety of improper functions occurring in contemporary investigations of partial differential equations, it seems necessary to widen the traditional concept of a mathematical function. The new concept, that of a “generalized function,” is abstract and cannot reproduce all aspects of the older concept of a function. In particular, it is not possible to ascribe a definite value to a generalized function at a point. Nevertheless, we shall see that in some sense such generalized functions can be described. In particular, it makes perfectly good sense to say that δ(t), which is a generalized function, vanishes on any open interval not containing t = 0.

In this chapter we shall outline two theories of generalized functions. One, essentially algebraic in nature, is restricted to generalized functions on a half line; the other, more closely related to functional analysis, places less restrictions on the independent variable. We shall also mention briefly other theories of generalized functions.

DELTA FUNCTIONS AND OTHER GENERALIZED FUNCTIONS

1.2 The Delta Function

Since the delta function is the idealization of functions that vanish outside a short interval, it...