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Unformatted text preview: 29 Delta Functions This chapter introduces mathematical entities commonly known as “delta functions”. As we will see, a delta function is not really a function, at least not in the classical sense. Nonetheless, with a modicum of care, they can be treated like functions. More importantly, they are useful. They are valuable in modeling both “strong forces of brief duration” (such as the force of a baseball bat striking a ball) and ”point masses”. Moreover, their mathematical properties turn out to be remarkable, making them some of the simplest “functions” to deal with. After a little practice, you may rank them with the constant functions as some of your favorite functions to deal with. Indeed, the basic delta function has a relation with the constant function f ≡ 1 that will allow us to expand our discussion of Duhamel’s principle. 29.1 Visualizing Delta Functions What is commonly called “the delta function”— traditionally denoted by δ( t ) — is best thought of as shorthand for a particular limiting process. One standard way to visualize δ( t ) is as the limit δ( t ) = lim ǫ → + 1 ǫ rect ( ,ǫ) ( t ) . Look at the function we are taking the limit of, 1 ǫ rect ( ,ǫ) ( t ) = if t < 1 ǫ if < t < ǫ if ǫ < t . Graphs of this for various small positive values of ǫ have been sketched in figure 29.1a. Notice that, for each ǫ , the nonzero part of the graph forms a rectangle of width ǫ and height 1 / ǫ . Consequently, the area of this rectangle is ǫ · 1 / ǫ = 1 . Keep in mind that we are taking a limit as ǫ → 0 ; so ǫ is “small”, which means that this rectangle is very narrow and very high, starts at t = 0 , and is of unit area. As we let ǫ → 0 this “very narrow and very high rectangle starting at t = 0 and of unit area” becomes an “infinitesimally narrow and infinitely high ‘spike’ at t = enclosing unit area”. Strictly speaking, there is no function whose graph is such a spike. The closest we can come is the function that is zero everywhere except at t = 0 , where we pretend the function is infinite. 581 582 Delta Functions (a) (b) 1 α α + ǫ 1 1 ǫ = 1 ǫ = 1 ǫ = 1 2 ǫ = 1 2 ǫ = 1 4 ǫ = 1 4 Figure 29.1: The graphs of (a) 1 ǫ rect ( ,ǫ) ( t ) and (b) 1 ǫ rect ( ,ǫ) ( t − α) (equivalently 1 ǫ rect (α,α + ǫ) ( t ) ) for ǫ = 1 , ǫ = 1 / 2 and ǫ = 1 / 4 . This sort of gives the infinite spike, but the “area enclosed” is not at all well defined. Still, the visualization of the delta function as “an infinite spike enclosing unit area” is useful, just as it is useful in physics to sometimes pretend that we can have a “point mass” (an infinitesimally small particle of nonzero mass)....
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 Summer '09
 Math, Differential Equations, Equations, Impulse response, Dirac delta function, Delta Functions

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