Page 100 - Advanced engineering mathematics

80 CHAPTER 3 The Laplace Transform

12. Show that 19. f has the graph of Figure 3.3.
(n+1)T T
e −st f (t)dt = e −nsT e −st f (t)dt. f(t)
nT 0
13. From Problems 11 and 12, show that E sin(ωt)

∞ T E

L[ f ](s) = e −nsT e −st f (t)dt.
0 t
n=0
π/ω 2π/ω 3π/ω
14. Recall the geometric series
FIGURE 3.3 Function for Problem 19, Section 3.1.
∞
1
n
r =
1 −r
n=0 20. f has the graph of Figure 3.4.
for |r| < 1. With this and the result of Problem 13,
show that f(t)
1 T
L[ f ](s) = e −st f (t)dt.
1 − e −sT 0
3
In each of Problems 15 through 22, a periodic function is
given (sometimes by a graph). Use the result of Problem
0 t
14 to compute its Laplace transform.
0 2 8 10 16 18 24
15. f has period of 6, and FIGURE 3.4 Function for Problem 20, Section 3.1.

5for 0 < t ≤ 3,
f (t) =
0for 3 < t ≤ 6 21. f has the graph of Figure 3.5.
16. f (t) =|E sin(ωt)| with E and ω positive numbers. f(t)
17. f has the graph of Figure 3.1.
h
f(t)

5 t
t 0 a 2a 3a 4a 5a
5 10 30 35 55 60
FIGURE 3.5 Function for Problem 21, Section 3.1.
FIGURE 3.1 Function for Problem 17, Section 3.1.
22. f has the graph of Figure 3.6.
18. f has the graph of Figure 3.2.
f(t)
f(t)
h

t
t
0 a 2a 3a 4a 5a 6a
0 6 12 18
FIGURE 3.6 Function for Problem 22,
FIGURE 3.2 Function for Problem 18, Section 3.1.
Section 3.1.

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October 14, 2010 14:14 THM/NEIL Page-80 27410_03_ch03_p77-120