Page 449 - Advanced engineering mathematics

13.2 The Fourier Series of a Function 429

SECTION 13.1 PROBLEMS

1. Let 2. Let p(x) be a polynomial. Prove that there is no number
N k such that p(x) = k sin(nx) on [0,π] for any positive
4 1 − (−1) n
S N (x) = sin(nx). integer n.
π n 3
n=1
3. Let p(x) be a polynomial. Prove that there is no finite
Construct graphs of S N (x) and x(π − x),for 0≤ x ≤π, N
sum b n sin(nx) that is equal to p(x) for 0≤ x ≤π,
for N =2andthen N =10. This will give some sense of n=1
for any choice of the numbers b 1 ,··· ,b N .
the correctness of Fourier’s claim that this polynomial
could be exactly represented by the infinite series
∞
4 1 − (−1) n
sin(nx)
π n 3
n=1
on [0,π].
13.2 The Fourier Series of a Function

Let f (x) be defined on [−L, L]. We want to choose numbers a 0 ,a 1 ,a 2 ··· and b 1 ,b 2 ,···
so that
∞
1
f (x) = a 0 + [a k cos(kπx/L) + b k sin(kπx/L)]. (13.1)
2
k=1

This is a decomposition of the function into a sum of terms, each representing the influence of a
different fundamental frequency on the behavior of the function.
To determine a 0 , integrate equation (13.1) term by term to get
1
L L
f (x)dx = a 0 dx
2
−L −L
L L
∞

+ a k cos(kπx/L)dx + b k sin(kπx/L)dx
k=1 −L −L
1
= a 0 (2L) = πa 0 .
2
because all of the integrals in the summation are zero. Then
1 L
a 0 = f (x)dx. (13.2)
L −L
To solve for the other coefficients in the proposed equation (13.1), we will use the following three
facts, which follow by routine integrations. Let m and n be integers. Then

L
cos(nπx/L)sin(mπx/L)dx = 0. (13.3)
−L
Furthermore, if n = m, then

L L
cos(nπx/L)cos(mπx/L)dx = sin(nπx/L)sin(mπx/L)dx = 0. (13.4)
−L −L

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 14:57 THM/NEIL Page-429 27410_13_ch13_p425-464