Page 860 - Advanced_Engineering_Mathematics o'neil

840 Answers to Selected Problems

Section 13.5 The Phase Angle Form
1. αf (t + p) + β fg(t + p) = αf (t) + βg(t)
3. f (t + p + h) − f (t + p)
f (t + p)= lim h→0
h
f (t + h) − f (t)

= lim h→0 = f (t)
h
5. The Fourier series is
∞
1 2 1
+ sin((2n − 1)πx),
2 π 2n − 1
n=1
so the phase angle form is
∞
2 1 π
1 + cos (2n − 1)πx −
π 2n − 1 2
n=1
7.
∞
19 1 nπx
+ c n cos + δ n
8 π 2 2
n=1
where
1
2
2
2
2
c n = 8 + 5n π − 12nπ sin(3nπ/2) + 4(n π − 2)cos(3nπ/2)
n 2
and

sin(3nπ/2) − nπ/2 − 2cos(3nπ/2)
δ n =−arctan .
nπ sin(3nπ/2) + cos(3nπ/2) − 1
9. Write f (x) = x for 0 ≤ x < 1and f (x) = x − 2for1 < x < 2. The phase angle form of the Fourier series is
∞
2 1 π
cos nπx + (−1) n .
π n 2
n=1
11. Write f (x) = 1for0 ≤ x < 1and f (x) = 2for1 < x < 3. The phase angle form of the Fourier series is
∞
3 2 1 πx π n
+ cos (2n − 1) + (1 − (−1) ) .
2 π 2n − 1 2 2
n=1
Section 13.6 Complex Fourier Series
3i ∞ 1
1. 3 + + e 2nπix/3
π n=−∞,n
=0 n
3 1 1
3. − ∞ (sin(nπ/2) + (cos(nπ/2) − 1)i)e nπix/2
4 2π n=−∞,n
=0 n
1 3i
5. + ∞ e (2n−1)πix/2
n=−∞,n
=0
2 π
Section 13.7 Filtering of Signals
1. The complex Fourier series is
∞
i

n
((−1) − 1)e inπt/2 .
nπ
n=−∞,n
=0
The Nth partial sum is
∞
4 1 (2n − 1)πt
S N (t) = sin .
π 2n − 1 2
n=1
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October 14, 2010 17:50 THM/NEIL Page-840 27410_25_Ans_p801-866