usinagem cnc

MA64A - Solu¸˜es Lista 2 co 8
16 + ω 2
4i
πω
ˆ
(b) f (ω) = − sin2 ω 2

ˆ
1. (a) f (ω) =

= −2i

1 − cos(πω) ω 2 sin(ω − 3)
ˆ
2. (a) f (ω) = ω−3 2 sin(2ω) −i 3 ω
ˆ
(b) f (ω) = e 2 ω 2 sin(2ω) 2 sin(ω)
ˆ
(c) f (ω) =
−
ω ω √ −ω2
ˆ
(d) f (ω) = −iω πe 4
−iωπ −3|w|
ˆ
(e) f (ω) = e 6
ˆ
(f) f (ω) = 2πe−|ω| e−2iω π ˆ
(g) f (ω) = [P4 (ω − πx) + P4 (ω + πx)]
8
1
ˆ
(h) f (ω) =
(1 + iω)2 + 1
ˆ
(i) f (ω) = 6πδ(ω) + 2πδ(ω + 3)
ˆ
(j) f (ω) = π[δ(ω − 3) + δ(ω + 3)] + e−i4ω
ˆ
3. (a) f (ω) = −4

ω cos(ω) − sin(w) ω3 ∞

(b)

x cos x − sin x x 3π cos dx = − . Sugest˜o: Use o Teorema de reprea
3
x
2
16
0
senta¸ao para f aplicado em x = 1 . Note que f ´ uma fun¸ao par. c˜ e c˜ 2
∞

dx π = . Sugest˜o: Use a identidade de Parseval para a Transfora
2 2
2
−∞ (1 + x ) mada de Fourier.
∞
cos(3x) dx = πe−3 . Sugest˜o: Escriba a transformada de Fourier de a (b)
1 + x2
−∞
1 f (x) = 1+x2 notando que f ´ uma fun¸ao impar. e c˜

4. (a)

∞

5.

1 − cos(πw) π sin(πw) dw = . Sugest˜o: Usando a transformada de Fourier a w
4
0 de f (Exerc´ 1 b.) use o Teorema de representa¸ao aplicado em x = π. Note que ıcio c˜ f ´ uma fun¸ao impar. e c˜

1

2π sin(2x) x 2π sin2 (x)
(b) (f1 ∗ f2 )(x) = x2 π
(c) (f1 ∗ f2 )(x) =
1 + x2

6. (a) (f1 ∗ f2 )(x) =

1
7. (a) y(x) = Ce−x + e−x u(x) −
2
1 −x
(b) y(x) = Ce−x + e u(x) −
3
−x
(c) y(x) = Ce + e−x u(x)

1 −3x e u(x)
2
1 −4x e u(x)
3

1
1
(d) y(x) = C1 e−3x + C2 e−5x + P1 (x) ∗ e−3x u(x) − P1 (x) ∗ e−5x
2
2
1
1
(e) y(x) = C1 e−3x + C2 e−5x + P2 (3x) ∗ e−3x u(x) − P2 (3x) ∗ e−5x
2
2

2