Page 485 - Advanced engineering mathematics

THE FOURIER INTEGRAL FOURIER
COSINE AND SINE INTEGRALS THE
CHAPTER 14 FOURIER TRANSFORM FOURIER COSINE
AND SINE TRANSFORMS THE DISCRETE
The Fourier

Integral and

Transforms

14.1 The Fourier Integral

If f (x) is defined for −L ≤ x ≤ L, we may be able to represent f (x) as a Fourier series on this
interval. However, Fourier series are tied to intervals. If f is defined over the entire line and is
not periodic, then the idea of a Fourier series representation is replaced with the idea of a Fourier
integral representation, in which the role of ∞ is played by ∞ .
n=0 0
We will give an informal argument to suggest the form that the Fourier integral should take.
∞
Assume that f is absolutely integrable, which means that | f (x)|dx converges. We also
−∞
assume that f is piecewise smooth on every interval [−L, L].
Write the Fourier series of f (x) on an arbitrary interval [−L, L]. With the formulas for the
coefficients included, this series is

1 L 1 L
∞
f (ξ)dξ + f (ξ)cos(nπξ/L)dξ cos(nπx/L)
2L −L n=1 L −L
L
1
+ f (ξ)sin(nπξ/L)dξ sin(nπx/L) .
L −L
We want to let L →∞ to obtain a representation of f (x) over the entire real line. It is not clear
what this quantity approaches, if anything, as L →∞, so we will rewrite some terms. First,
let
nπ
ω n =
L
and
π
ω n − ω n−1 = = ω.
L

465

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 16:43 THM/NEIL Page-465 27410_14_ch14_p465-504