Page 510 - Advanced engineering mathematics

490 CHAPTER 14 The Fourier Integral and Transforms

1 for −4π ≤ t ≤ 4π, 1 for −2 ≤ t ≤ 2,
2
24. f (t) = cos(at), w(t) = 27. f (t) = (t + 2) , w(t) =
0 for |t| > 4π. 0 for |t| > 2.

1 for 0 ≤ t ≤ 4, 1 for 3π ≤ t ≤ 5π,
−t
25. f (t) = e , w(t) = 28. f (t) = H(t − π), w(t) =
0 for t < 0or t > 4. 0 for t < 3π or t > 5π.

1for −1 ≤ t ≤ 1,
t
26. f (t) = e sin(πt), w(t) =
0for |t| > 1.
14.4 Fourier Cosine and Sine Transforms

∞
If f is piecewise smooth on each interval [0, L] and | f (t)|dt converges, then at each t where
0
f is continuous, the Fourier cosine integral for f is
∞

f (t) = a ω cos(ωt)dω,
0
where
2 ∞
a ω = f (t)cos(ωt)dt.
π 0

We define the Fourier cosine transform of f by
∞

F C [ f ](ω) = f (t)cos(ωt)dt. (14.16)
0

ˆ
Often we denote F C [ f ](ω) = f C (ω).
Notice that
π
ˆ
f C (ω) = a ω
2
and that
2 ∞
ˆ
f (t) = f c (ω)cos(ωt)dω. (14.17)
π 0
Equations (14.16) and (14.17) form a transform pair for the Fourier cosine transform. Equa-
ˆ
tion (14.17) is the inverse Fourier cosine transform, reproducing f from f c . This inverse is
denoted f ˆ C −1 .

EXAMPLE 14.12
Let K be a positive number and let

1 for 0 ≤ t ≤ K
f (t) =
0 for t > K.
Then
∞ sin(Kω)
K
ˆ
f C (ω) = f (t)cos(ωt)dt = cos(ωt)dt = .
ω
0 0

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October 14, 2010 16:43 THM/NEIL Page-490 27410_14_ch14_p465-504