Complex Analysis - Maths KU

6.7 Applications 385
whenever both transforms exist. Use the linearity defined above along with the
definitions
sinh kt = ekt − e −kt
, cosh kt = ekt − e −kt
k a real constant, to find � {sinh kt} and � {cosh kt}.
2
8. State a condition on s that is sufficient to guarantee the existence of the Laplace
transforms in Problem 7.
In Problems 9–18, use the theory of residues to compute the inverse Laplace transform
� −1 {F (s)} for the given function F (s).
9. 1
s6 1
10.
(s − 5) 3
11.
1
s2 +4
12.
s
(s2 +1) 2
13.
1
s2 − 3
14.
1
(s − a) 2 + b2 15.
e −as
s2 , a > 0
− 5s +6
16.
e −as
, a > 0
(s − a) 2
17.
1
s4 − 1
18.
s +4
s2 +6s +11
In Problems⎧19 and 20, find the Fourier transform (19) of the given function.
⎨ 0, x ≤ 0
19. f(x) =
⎩ e −x ⎧
⎨ sin x, |x| ≤π
20. f(x) =
, x > 0
⎩ 0, |x| >π
21. Use the inverse Fourier transform (20) and the theory of residues to recover the
function f in Problem 19.
1
22. The Fourier transform of a function f is F (α) = . Use the inverse
(1 − iα) 2
Fourier transform (20) and the theory of residues to find the function f.
Focus on Concepts
23. For the result obtained in Problem 8, find values of γ that can be used in the
inverse transform (7).
24. (a) If F (α) is the Fourier transform of f(x), then the function |F (α)| is called
the amplitude spectrum of f. Find the amplitude spectrum of
⎧
⎨ 1, |x| ≤1
f(x) =
.
⎩ 0, |x| > 1
Graph |F (α)|.
(b) Do some additional reading and find an application of the concept of the
amplitude spectrum of a function.
⎧
⎨ x, 0