Page 862 - Advanced_Engineering_Mathematics o'neil

842 Answers to Selected Problems

9. For all x,
∞

2
cos(ωx)dω = e −|x| .
π(1 + ω )
2
0
11. Because f (t)cos(ω(t − x)) is even in ω and f (t)sin(ω(t − x)) is odd, we can write
1 ∞ ∞ 1 ∞ ∞
f (t)cos(ω(t − x))dt dω = f (t)cos(ω(t − x))dω dt
π 0 −∞ 2π −∞ −∞
1 ∞ ∞
= f (t)e iω(t−x) dω dt.
2π
−∞ −∞
Complete the derivation by carrying out the inner integration in the last double integral.
Section 14.2 Fourier Cosine and Sine Integrals
1. Sine integral:
4

∞
2
[10ω sin(10ω) − (50ω − 1)cos(10ω) − 1]sin(ωx)dω.
πω 3
0
Cosine integral:
∞
4 2

[10ω cos(10ω) − (50ω − 1)sin(10ω)]cos(ωx)dω.
πω 3
0
2
Both integrals converge to x for 0 ≤ x < 10, to 50 if x = 10andto0if x > 10.
3. Sine integral:
∞
2

[1 + cos(ω) − 2cos(4ω)]sin(ωx)dω.
πω
0
Cosine integral:

∞ 2
[2sin(4ω) − sin(ω)]cos(ωx)dω.
πω
0
Both integrals converge to 1 for 0 < x < 1, to 3/2for x = 1, to 2 for 1 < x < 4, to 1 for x = 4andto 0for x > 4. The
cosine integral converges to 1 at x = 0 while the sine integral converges to 0 at x = 0.
5. Sine integral:
∞
2 4

[1 + (1 − 2π)cos(πω) − 2cos(3πω)]+ sin(πω) sin(ωx)dω.
πω πω 2
0
Cosine integral:
∞ 2 4

[(2π − 1)sin(πω) + 2sin(3πω)]+ [cos(πω) − 1] cos(ωx)dω.
πω πω 2
0
Both integrals converge to 1 + 2x for 0 < x <π,to (3 + 2π)/2for x = π,to2for π< x < 3π,to1for x = 3π,andto
0for x > 3π. The sine integral converges to 0 at x = 0, while the cosine integral converges to 1 for x = 0.
7. Sine integral:

∞ 3
2 ω
sin(ωx)dω.
0 π 4 + ω 4
Cosine integral:
∞
2 2 + ω
2
cos(ωx)dω.
π 4 + ω 4
0
Both integrals converge to e −x cos(x) for x > 0. the cosine integral converges to 1 for x = 0 and the sine integral
convergesto0at x = 0.
9. Sine integral:
∞
2k

(1 − cos(cω))sin(ωx)dω.
πω
0
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October 14, 2010 17:50 THM/NEIL Page-842 27410_25_Ans_p801-866