Page 861 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 841

The Nth Cesàro sum is
N
4 2n − 1 1 (2n − 1)πt
σ N (t) = 1 − sin .
π N 2n − 1 2
n=1
3. The complex Fourier series is
∞
i nπ

n
(−1) − cos e inπt .
nπ 2
n=−∞
The Nth partial sum is
∞
2

n
S N (t) = (cos(nπ/2) − (−1) )sin(nπt)
nπ
n=1
The Nth Cesàro sum is
n 2 nπ
N

n
σ N (t) = 1 − cos − (−1) sin(nπt).
N nπ 2
n=1
5. The complex Fourier series is
∞
17 1 i
n
n
+ (1 − (−1) ) + (6(−1) − 5) e inπt/2 .
4 2n π 2 2nπ
2
n=−∞,n
=0
The Nth Cesàro sum is
N
17 1 |n|
n
σ N (t) = + 1 − (1 − (−1) )cos(nπt)
2
4 n π 2 N
n=1
N
i |n| n
+ 1 − (5 − 6(−1) )sin(nπt).
nπ N
n=1
7. The Nth partial sums are
CHAPTER FOURTEEN THE FOURIER INTEGRAL AND FOURIER TRANSFORMS
Section 14.1 The Fourier Integral
1. The Fourier integral is
∞ 2sin(πω) 2cos(πω)

− sin(ωx)dω,
πω 2 ω
0
converging to −π/2if x =−π,to x for −π< x <π,to π/2for x = π andto0if |x| >π.
3.
2
∞

(1 − cos(πω)) sin(ωx)dω,
πω
0
converging to −1/2at x =−π,to −1for −π< x < 0, to 0 if x = 0,to1if0 < x <π,to1/2if x = π and to 0 if
|x| >π.
5.
∞ 1

2
[400ω cos(100ω) + (20000ω − 4)sin(100ω)]cos(ωx)dω,
πω 3
0
2
converging to x if −100 < x < 100, to 5000 if x =±100, and to 0 if |x| > 100.
7.
2
2
4 ∞
cos(πω)(cos (πω) − 1) sin(πω)cos (πω)
cos(ωx) + sin(ωx) dω,
2
π 0 ω − 1 1 − ω 2
converging to sin(x) if −3π< x <π andto0if x < −3π or x ≥ π.
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October 14, 2010 17:50 THM/NEIL Page-841 27410_25_Ans_p801-866