Séries de Fourier

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Universidade Te
nológi
a Federal do Paraná
Departamento A
adêmi
o de Matemáti
a
Cál
ulo 4 (MA64A & MA74C)
Prof. Roy Wilhelm Probst
Lista 1 - Séries de Fourier
1. f(x) = x2, 0 < x < 2pi, f(x + 2pi) = f(x).
2. f(x) = x + pi, −pi < x < pi, f(x + 2pi) = f(x).
3. f(x) = x, −2 < x < 2, f(x + 4) = f(x).
4. f(x) =


−
4
pi
x− 2, se −
pi
2
< x < 0
−
4
pi
x + 2, se 0 < x <
pi
2
, f(x + pi) = f(x).
5. f(x) =


3, se − 2 < x ≤ −1
−3x, se − 1 ≤ x ≤ 1
−3, se 1 ≤ x < 2
, f(x + 4) = f(x).
6. f(x) =


2, se − 2 ≤ x ≤ −1
−2x, se − 1 ≤ x ≤ 0
2x, se 0 ≤ x ≤ 1
2, se 1 ≤ x ≤ 2
, f(x + 4) = f(x).
Respostas
1. f(x) =
4pi2
3
+ 4
∞∑
n=1
[
1
n2
cos(nx)−
pi
n
sin(nx)
]
2. f(x) = pi + 2
∞∑
n=1
(−1)n+1
n
sin(nx)
3. f(x) =
4
pi
∞∑
n=1
(−1)n+1
n
sin
(npi
2
x
)
4. f(x) =
4
pi
∞∑
n=1
1
n
sin(2nx)
5. f(x) =
6
pi
∞∑
n=1
(−1)n −
2
npi
sin
(npi
2
)
n
sin
(npi
2
x
)
6. f(x) =
3
2
+
8
pi2
∞∑
n=1
cos
(npi
2
)
− 1
n2
cos
(npi
2
x
)