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470 CHAPTER 14 The Fourier Integral and Transforms

Next compute the sine coefficients

2 ∞ 2 ω
B ω = e −kξ sin(ωξ)dξ = .
2
π 0 π k + ω 2
Then, for x > 0, we also have
2 ∞ ω
−kx
e = sin(ωx)dω.
2
π 0 k + ω 2
However, this integral is zero for x = 0 and so does not represent f (x) there.
These integral representations are called Laplace’s integrals because A ω is 2/π times the
Laplace transform of sin(kx), while B ω is 2/π times the Laplace transform of cos(kx).

SECTION 14.2 PROBLEMS

⎧
In each of Problems 1 through 10, find the Fourier ⎪2x + 1for 0 ≤ x ≤ π
⎨
cosine and sine integral representations of the func- 5. f (x) = 2 for π< x ≤ 3π
tion. Determine what each integral representation ⎪ 0 for x > 3π.
⎩
converges to.
⎧
⎪x for 0 ≤ x ≤ 1
⎨

6. f (x) = x + 1for 1 < x ≤ 2
x 2 for 0 ≤ x ≤ 10
1. f (x) = ⎪ 0 for x > 2
⎩
0 for x > 10
7. f (x) = e −x cos(x) for x ≥ 0

−3x
sin(x) for 0 ≤ x ≤ 2π 8. f (x) = xe for x ≥ 0
2. f (x) = 9. Let k be a nonzero number and c a positive number,
0 for x > 2π
and

⎧
⎪1for 0 ≤ x ≤ 1 k for 0 ≤ x ≤ c
f (x) =
⎨
3. f (x) = 2for 1 < x ≤ 4 0 for x > c.
⎪
0for x > 4
⎩
10. f (x) = e −2x cos(x) for x ≥ 0.
11. Use the Laplace integrals to compute the Fourier

cosh(x) for 0 ≤ x ≤ 5 cosine integral of f (x) = 1/(1 + x ) and the Fourier
2
4. f (x) =
2
0 for x > 5 sine integral of g(x) = x/(1 + x ).
14.3 The Fourier Transform
We will use equation (14.4) to derive a complex form of the Fourier integral representation of a
function, and then use this to define the Fourier transform.
Suppose f is absolutely integrable on the real line, and piecewise smooth on each [−L, L].
Then, at any x,
1 1 ∞ ∞
( f (x+) + f (x−)) = f (ξ)cos(ω(ξ − x))dξ dω.
2 π 0 −∞
Recall that
1 ix −ix
cos(x) = e + e
2

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October 14, 2010 16:43 THM/NEIL Page-470 27410_14_ch14_p465-504