Page 863 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 843

Cosine integral:
∞
2k

sin(cω)cos(ωx)dω.
πω
0
Both integrals converge to k for 0 < x < c,to k/2for x = c, and to 0 for x > c, while the sine integral converges to 0 at
x = 0, and the cosine integral to k there.
11. From the Laplace integrals and the convergence theorem, for x > 0,
2k ∞ 1 2 ∞ ω
e −kx = cos(ωx)dω = sin(ωx)dω.
2
π 0 k + ω 2 π 0 k + ω 2
2
Put k = 1 and interchange the symbols x and ω to obtain
πe −ω ∞ 1
A ω = = cos(ωx)dx
2k 0 1 + x 2
and
πe −ω ∞ x
B ω = = sin(ωx)dx.
2 0 1 + x 2
2
Therefore, the Fourier cosine integral of 1/(1 + x ) is
∞ 1

C(x) = e −ω cos(ωx)dω = 2 for x ≥ 0.
0 1 + x
2
And the Fourier sine integral of x/(1 + x ) is
∞ x

S(x) = e −ω sin(ωx)dω = 2 for x > 0.
0 1 + x
Section 14.3 The Fourier Transform
1. 2i[cos(ω) − 1]/ω 3. 10e −7iω sin(4ω)/ω
4
5. e −(1+4iω)k/4 7. πe −|ω|
1 + 4iω
24
9. e 2iω
16 + ω 2

2
11. 18 e −8t 2 e −4it 13. H(t + 2)e −10−(5−3i)t
π
15. H(t)[2e −3t − e −2t ] 17. H(t)te −t
1
2
2
∞ 1 ∞
19. | f (t)| dt = ˆ f (ω) ˆ f (ω)dω = | ˆ f (ω)| dω
2π 2π
−∞ −∞
21. 3π
3 2
23. (2/ω )[25ω sin(5ω) + 10ω cos(5ω) − 2sin(5ω)]
1 i
−4
−4
−4
−4
25. (1 − e cos(4ω) + e sin(4ω)) + (e sin(4ω) + (e cos(4ω) − 1)ω)
1 + ω 2 1 + ω 2
4 8i
27. (sin(2ω)(4ω − 1) + 2ω cos(2ω)) + (2ω cos(2ω) − sin(2ω))
2
ω 3 ω 2
Section 14.4 Fourier Cosine and Sine Transforms
1 ω
1. ˆ f C (ω) = , ˆ f S (ω) =
1 + ω 2 1 + ω 2
3. For ω
=±1,
1 sin(K(1 − ω)) sin(K(1 + ω))

ˆ f C (ω) = + ,
2 1 − ω 1 + ω
K 1
ˆ f C (−1) = ˆ f C (1) = + sin(2K)
2 2
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October 14, 2010 17:50 THM/NEIL Page-843 27410_25_Ans_p801-866