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Unformatted text preview: 28 Piecewise-Defined Functions and Periodic Functions At the start of our study of the Laplace transform, it was claimed that the Laplace transform is particularly useful when dealing with nonhomogeneous equations in which the forcing func- tions are not continuous. Thus far, however, weve done precious little with any discontinuous functions other than step functions. Let us now rectify the situation by looking at the sort of discontinuous functions (and, more generally, piecewise-defined functions) that often arise in applications, and develop tools and skills for dealing with these functions. We will also take a brief look at transforms of periodic functions other than sines and cosines. As you will see, many of these functions are, themselves, piecewise defined. And finally, we will use some of the material weve recently developed to re-examine the issue of resonance in mass/spring systems. 28.1 Piecewise-Defined Functions Piecewise-Defined Functions, Defined When we talk about a discontinuous function f in the context of Laplace transforms, we usually mean f is a piecewise continuous function that is not continuous on the interval ( , ) . Such a function will have jump discontinuities at isolated points in this interval. Computationally, however, the real issue is often not so much whether there is a nonzero jump in the graph of f at a point t , but whether the formula for computing f ( t ) is the same on either side of t . So we really should be looking at the more general class of piecewise-defined functions that, at worst, have jump discontinuities. Just what is a piecewise-defined function? It is any function given by different formulas on different intervals. For example, f ( t ) = if t < 1 1 if 1 < t < 2 if 2 < t and g ( t ) = if t 1 t 1 if 1 < t < 2 1 if 2 t are two relatively simple piecewise-defined functions. The first (sketched in figure 28.1a) is discontinuous because it has nontrivial jumps at t = 1 and t = 2 . However, the second 547 548 Piecewise-Defined and Periodic Functions (a) (b) T T 1 1 1 1 2 2 Figure 28.1: The graphs of two piecewise-defined functions. function (sketched in figure 28.1b) is continuous because t 1 goes from 0 to 1 as t goes from 1 to 2 . There are no jumps in the graph of g . By the way, we may occasionally refer to the sort of lists used above to define f ( t ) and g ( t ) as conditional sets of formulas or sets of conditional formulas for f and g , simply because these are sets of formulas with conditions stating when each formula is to be used. Do note that, in the above formula set for f , we did not specify the values of f ( t ) when t = 1 or t = 2 . This was because f has jump discontinuities at these points and, as we agreed in chapter 24 (see page 492), we are not concerned with the precise value of a function at its discontinuities. On the other hand, using the formula set given above for g...
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