Page 472 - Advanced engineering mathematics

452 CHAPTER 13 Fourier Series

SECTION 13.4 PROBLEMS

1. Let 5. Let f and f be piecewise continuous on [−L, L].Use

Bessel’s inequality to show that
0 for −π ≤ x ≤ 0
f (x) =
x for 0 < x ≤ π. L nπx
lim f (x)cos dx
n→∞ −L L
(a) Write the Fourier series of f (x) on [−π,π] and L nπx
show that this series converges to f (x) on (−π,π). = lim f (x)sin dx = 0.
n→∞ −L L
(b) Use Theorem 12.5 to show that this series can be
This result is called Riemann’s lemma.
integrated term by term.
6. Prove Theorem 13.8 by filling in the details of the fol-
(c) Use the results of (a) and (b) to obtain a trigono-
metric series expansion of x f (t)dt on [−π,π]. lowing argument. Denote the Fourier coefficients of
−π
f (x) by lower case letters, and those of f (x) by upper
2. Let f (x) =|x| for −1 ≤ x ≤ 1. case. Show that
(a) Write the Fourier series for f on [−1,1]. A 0 = 0, A n = nπ b n , and B n =− nπ a n .
L L
(b) Show that this series can be differentiated term by
term to yield the Fourier expansion of f (x) on Show that

[−π,π]. 2 1
2
0 ≤ A − |A n |+

(c) Determine f (x) and expand this function in a n n n 2
Fourier series on [−π,π]. Compare this result with
for n = 1,2,···, with a similar inequality for B n .Add
that of (b).
these two inequalities to obtain
3. Let f (x) = x sin(x) for −π ≤ x ≤ π.
1 1 1
2
2
(|A n |+|B n |) ≤ (A + B ) + .
n
n
(a) Write the Fourier series for f on [−π,π] n 2 n 2
(b) Show that this series can be differentiated term Hence show that
by term and use this fact to obtain the Fourier
L L
2
2
expansion of sin(x) + x cos(x) on [−π,π]. |a n |+|b n |≤ (A + B ) + .
n
n
2
2π π(n )
(c) Write the Fourier series for sin(x) + x cos(x) on
Thus show by comparison that
[−π,π] and compare this result with that of (b).
∞
2
4. Let f (x) = x for −3 ≤ x ≤ 3.
(|a n |+|b n |)
(a) Write the Fourier series for f on [−3,3]. n=1
(b) Show that this series can be differentiated term by converges. Finally, show that
term and use this to obtain the Fourier expansion
of 2x on [−3,3]. |a n cos(nπx/L) + b n sin(nπx/L)|≤|a n |+|b n |
(c) Expand 2x in a Fourier series on [−3,3] and and apply a theorem of Weierstrass on uniform conver-
compare this result with that of (b). gence.
13.5 Phase Angle Form
A function f has period p if f (x + p) = f (x) for all x. The smallest positive p for which
this holds is the fundamental period of f . For example sin(x) has fundamental period 2π.
The graph of a function with fundamental period p simply repeats itself over intervals of
length p. We can draw the graph for −p/2 ≤ x < p/2, then replicate this graph on p/2 ≤ x <
3p/2, 3p/2 ≤ x < 5p/2, −3p/2 ≤ x < −p/2, and so on.

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October 14, 2010 14:57 THM/NEIL Page-452 27410_13_ch13_p425-464