Page 372 - Schaum's Outline of Theory and Problems of Advanced Calculus

Fourier Integrals

Fourier integrals are generalizations of Fourier series. The series representation
1
X n n x n x o
a 0
a n cos þ b n sin of a function is a periodic form on 1 < x < 1 obtained by gen-
2 þ L L
n¼1
erating the coefficients from the function’s definition on the least period ½ L; LŠ.Ifa function defined
on the set of all real numbers has no period, then an analogy to Fourier integrals can be envisioned as
letting L !1 and replacing the integer valued index, n,bya real valued function . The coefficients a n
and b n then take the form Að Þ and Bð Þ. This mode of thought leads to the following definition. (See
Problem 14.8.)

THE FOURIER INTEGRAL
Let us assume the following conditions on f ðxÞ:
1. f ðxÞ satisfies the Dirichlet conditions (Page 337) in every finite interval ð L; LÞ.
ð
1
2. j f ðxÞj dx converges, i.e. f ðxÞ is absolutely integrable in ð 1; 1Þ.
1
Then Fourier’s integral theorem states that the Fourier integral of a function f is
ð
1
fAð Þ cos x þ Bð Þ sin xg d
f ðxÞ¼ ð1Þ
0
ð
1
8 1
> f ðxÞ cos xdx
> Að Þ¼
<
where 1 (2)
1 ð 1
>
> f ðxÞ sin xdx
:
Bð Þ¼ 1
Að Þ and Bð Þ with 1 < < 1 are generalizations of the Fourier coefficients a n and b n . The
right-hand side of (1)is also called a Fourier integral expansion of f . (Since Fourier integrals are
improper integrals, a review of Chapter 12 is a prerequisite to the study of this chapter.) The result
(1) holds if x is a point of continuity of f ðxÞ.If x is a point of discontinuity, we must replace f ðxÞ by
as in the case of Fourier series. Note that the above conditions are sufficient but not
f ðx þ 0Þþ f ðx 0Þ
2
necessary.
363
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