Page 462 - Advanced engineering mathematics

442 CHAPTER 13 Fourier Series

for n = 0,1,2,···. Notice that we can compute a n strictly in terms of f on [0, L]. The construc-
tion of g showed us how to obtain this cosine series for f , but we do not need g to compute the
coefficients of this series.

Based on these ideas, define the Fourier cosine coefficients of f on [0, L] to be the numbers
2 L
a n = f (x)cos(nπx/L)dx (13.10)
L 0
for n = 0,1,2,···.The Fourier cosine series for f on [0, L] is the series

∞
1
a 0 + a n cos(nπx/L) (13.11)
2
n=1

in which the a s are the Fourier cosine coefficients of f on [0, L].
n
By applying Theorem 13.1 to g, we obtain the following convergence theorem for cosine
series on [0, L].

THEOREM 13.2 Convergence of Fourier Cosine Series
Let f be piecewise smooth on [0, L]. Then

1. If 0 < x < L, the Fourier cosine series for f on [0, L] converges to

1
( f (x+) + f (x−)).
2
2. At 0 this cosine series converges to f (0+).
3. At L this cosine series converges to f (L−).

EXAMPLE 13.9
2x
Let f (x) = e for 0 ≤ x ≤ 1. We will write the cosine expansion of f on [0,1]. The coefficients
are
1
2
2x
a 0 = 2 e dx = e − 1
0
and for n = 1,2,···,

1
2x
a n = 2 e cos(nπx)dx
0
1
4e cos(nπx) + 2nπe sin(nπx)
2x 2x
=
4 + n π 2
2
0
n
2
e (−1) − 1
= 4 .
2
4 + n π 2
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-442 27410_13_ch13_p425-464