Page 354 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 13] FOURIER SERIES 345

(a) Multiplying
X n x n x
1
a n cos þ b n sin
L L
f ðxÞ¼ A þ ð1Þ
n¼1
m x
by cos and integrating from L to L,using Problem 13.3, we have
L
ð L m x ð L m x
f ðxÞ cos dx ¼ A cos dx
L L L L
ð ð
1 L L
X m x n x m x n x
cos cos cos sin dx
a n dx þ b n
þ L L L L
n¼1 L L
¼ a m L if m 6¼ 0
1 ð L m x
Thus a m ¼ f ðxÞ cos dx if m ¼ 1; 2; 3; .. .
L L L
m x
(b) Multiplying (1)by sin and integrating from L to L,using Problem 13.3, we have
L
ð L m x ð L m x
f ðxÞ sin dx ¼ A sin dx
L L L L
ð L ð L
1
X m x n x m x n x
sin cos sin sin dx
a n dx þ b n
þ L L L L
n¼1 L L
¼ b m L
1 ð L m x
Thus b m ¼ f ðxÞ sin dx if m ¼ 1; 2; 3; .. .
L L L
(c) Integrating of (1) from L to L,using Problem 13.2, gives
L 1 L
ð ð
f ðxÞ dx ¼ 2AL or A ¼ f ðxÞ dx
L 2L L
1 ð L a 0
.
Putting m ¼ 0inthe result of part (a), we find a 0 ¼ f ðxÞ dx and so A ¼
L L 2
The above results also hold when the integration limits L; L are replaced by c; c þ 2L:
Note that in all parts above, interchange of summation and integration is valid because the series is
assumed to converge uniformly to f ðxÞ in ð L; LÞ. Even when this assumption is not warranted, the
coefficients a m and b m as obtained above are called Fourier coefficients corresponding to f ðxÞ, and the
corresponding series with these values of a m and b m is called the Fourier series corresponding to f ðxÞ.
An important problem in this case is to investigate conditions under which this series actually converges
to f ðxÞ. Sufficient conditions for this convergence are the Dirichlet conditions established in Problems
13.18 through 13.23.

13.5. (a) Find the Fourier coefficients corresponding to the function

0 5 < x < 0
3 0 < x < 5
f ðxÞ¼ Period ¼ 10
(b) Write the corresponding Fourier series.
(c) How should f ðxÞ be defined at x ¼ 5; x ¼ 0; and x ¼ 5in order that the Fourier series will
converge to f ðxÞ for 5 @ x @ 5?
The graph of f ðxÞ is shown in Fig. 13-6.