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468 CHAPTER 14 The Fourier Integral and Transforms

⎧ ⎧
⎪sin(x) for −4 ≤ x ≤ 0 ⎪1/2 for −5 ≤ x < 1
⎨ ⎨
4. f (x) = cos(x) for 0 < x ≤ 4 8. f (x) = 1 for 1 ≤ x ≤ 5
⎪ ⎪
0 for |x| > 4 0 for |x| > 5
⎩ ⎩

−|x|
x 2 for −100 ≤ x ≤ 100 9. f (x) = e
5. f (x) = 10. f (x) = xe −4|x|
0 for |x| > 100
11. Show that the Fourier integral of f (x) can be written

|x| for −π ≤ x ≤ 2π 1 ∞ sin(ω(t − x))
6. f (x) = lim f (t) dt.
0 for x < −π and for x > 2π ω→∞ π t − x
−∞

sin(x) for −3π ≤ x ≤ π
7. f (x) =
0 for x < −3π and for x >π
14.2 Fourier Cosine and Sine Integrals

We can define Fourier cosine and sine integral expansions for functions defined on the half-line
in a manner completely analogous to Fourier cosine and sine expansions of functions defined on
a half interval.
Suppose f (x) is defined for x ≥ 0. Extend f to an even function f e on the real line. where

f (x) for x ≥ 0,
f e (x) =
f (−x) for x < 0.
This reflects the graph of f (x) for x ≥ 0 back across the vertical axis to a function f e defined on
the entire line. Because f e is an even function, its Fourier coefficients are

1 ∞
A ω = f e (ξ)cos(ωξ)dξ
π
−∞
2 ∞
= f (ξ)cos(ωξ)dξ
π 0
and
1 ∞
B ω = f e (ξ)cos(ωξ)dξ = 0.
π
−∞
The Fourier integral of f e (x) contains only cosine terms. Since f e (x) = f (x) for x ≥ 0, this
expansion may be thought of as a cosine expansion of f (x), on the half-line x ≥ 0.

This leads us to define the Fourier cosine integral of f (x) on x ≥ 0tobe
∞

A ω cos(ωx)dω (14.5)
0
in which
2 ∞
A ω = f (ξ)cos(ωξ)dξ (14.6)
π 0
is the Fourier integral cosine coefficient.

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October 14, 2010 16:43 THM/NEIL Page-468 27410_14_ch14_p465-504