# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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I — = lim I — = lim ln A.

J- t A^to J - t A^to

Since lim ln A = þ, the improper integral diverges.

-The Laplace transform is named for the eminent French mathematician P. S. Laplace, who studied the relation (2) in 1782. However, the techniques described in this chapter were not developed until a century or more later. They are due mainly to Oliver Heaviside (1850-1925), an innovative but unconventional English electrical engineer, who made significant contributions to the development and application of electromagnetic theory.

6.1 Definition of the Laplace Transform 295

Let f(t) = t—p, t > 1, where p is a real constant and p = 1; the case p = 1 was EXAMPLE considered in Example 2. Then

3

ðÆ pA 1

t~p dt = lim t~p dt = lim -------(A1-p — 1).

J1 A^TO J1 A^TO 1 — p

/Æ

t— p dt converges for p > 1, but (incorporating the result of Example 2) diverges for p < 1. These

Æ

results are analogous to those for the infinite series ^ n—p.

n= 1

/> Æ

Before discussing the possible existence of / f (t) dt, it is helpful to define certain terms. A function f is said to be piecewise continuous on an interval à < t < â if the interval can be partitioned by a finite number of points à = t0 < ^ < ••• < tn = â so that

1. f is continuous on each open subinterval ti—1 < t < ti.

2. f approaches a finite limit as the endpoints of each subinterval are approached from within the subinterval.

In other words, f is piecewise continuous on à < t < â if it is continuous there except for a finite number of jump discontinuities. If f is piecewise continuous on à < t < â for every â > à, then f is said to be piecewise continuous on t > à. An example of a piecewise continuous function is shown in Figure 6.1.1.

If f is piecewise continuous on the interval a < t < A, then it can be shown that

J.A ë A

f ( t) dt exists. Hence, if f is piecewise continuous for t > a, then / f ( t) dt exists

a J a

for each A > a. However, piecewise continuity is not enough to ensure convergence

Æ

of the improper integral / f (t) dt, as the preceding examples show.

a

If f cannot bae integrated easily in terms of elementary functions, the definition of

Æ

convergence of / f (t) dt may be difficult to apply. Frequently, the most convenient

way to test the convergence or divergence of an improper integral is by the following comparison theorem, which is analogous to a similar theorem for infinite series.

FIGURE 6.1.1 A piecewise continuous function.

296

Chapter 6. The Laplace Transform

Theorem 6.1.1

Theorem 6.1.2

If f is piecewise continuous for t > a, if I f (t)| < g(t) when t > Mfor some positive

J-TO ¥* TO

g(t) dt converges, then / f(t) dt also converges. On

M J a

J. TO

g(t) dt diverges, then

M

´* TO

f(t) dt also diverges.

The proof of this result from the calculus will not be given here. It is made plausible,

TOTO

however, by comparing the areas represented by J g(t) dt and J | f (t)| dt. The

functions most useful for comparison purposes are ect and t-p, which were considered in Examples 1, 2, and 3.

We now return to a consideration of the Laplace transform L{ f (t)} or F(s), which is defined by Eq. (2) whenever this improper integral converges. In general, the parameter s may be complex, but for our discussion we need consider only real values of s. The foregoing discussion of integrals indicates that the Laplace transform F of a function f exists if f satisfies certain conditions, such as those stated in the following theorem.

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