Page 122 - Advanced engineering mathematics

102 CHAPTER 3 The Laplace Transform

−t
t
19. f (t) = e + f (t − τ)dτ Show that
0

t ∞
20. f (t) =−1 + t − 2 f (t − τ)sin(τ)dτ
0 F(s)G(s) = L[H(t − τ) f (t − τ)](s)g(τ)dτ.

t
21, f (t) = 3 + f (τ)cos(2(t − τ))dτ 0
0
2τ
22. f (t) = cos(t) + e −2t
t f (τ)e dτ Use the definitions of the Heaviside function and of
0
the transform to obtain
23. Solve for the replacement function r(t) if f (t) = A,
∞ ∞

constant, and m(t) = e −kt with k a positive constant. F(s)G(s) = e −st g(τ) f (t − τ)dτ.
Graph r(t). 0 τ
24. Solve for the replacement function r(t) if f (t) = A + Reverse the order of integration to obtain
Bt and m(t) = e −kt .Graph r(t).
∞ t −st

25. Solve for the replacement function r(t) if f (t) = A + F(s)G(s) = e g(τ) f (t − τ)dτ dt
0 0
Bt + Ct and m(t) = e −kt .Graph r(t).
2
∞

26. Prove the convolution theorem. Hint: First write = e −st ( f ∗ g)(t)dt.
0
∞

F(s)G(s) = F(s)e −sτ g(τ)dτ. From this, show that L[ f ∗ g](s) = F(s)G(s).
0
3.5 Impulses and the Delta Function
Informally, an impulse is a force of extremely large magnitude applied over an extremely
short period of time (imagine hitting your thumb with a hammer). We can model this idea
as follows. First, for any positive number consider the pulse δ defined by
1
δ (t) = [H(t) − H(t − )].

This pulse, which is graphed in Figure 3.22, has magnitude (height) of 1/ and dura-
tion of .The Dirac delta function is thought of as a pulse of infinite magnitude over an
infinitely short duration and is defined to be

δ(t) = lim δ (t).
→0+

δ (t)
ε
1/ε

ε t

FIGURE 3.22 δ (t)

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October 14, 2010 14:14 THM/NEIL Page-102 27410_03_ch03_p77-120