MATH 267
Due: Feb 15, 2012, in the class
ASSIGNMENT # 5
You have FIVE problems to hand-in. Hand in written solutions for grading at the BEGINNING of the
lecture on the due date. Illegible, disorganized or partial solutions will receive no credit.
*Staple your HW. From now on, you will get F IV E marks OFF if you do not staple your HW!
Note that the instructor will NOT provide stapler.
1. (a) Let f (t) = |t| for −2 ≤ t ≤ 2. Let F (t) be a periodic extension of f with period 4.
i. Sketch the graph of F (t) (show at least three periods).
ii. Write down the complex Fourier series of F (t).
iii. Write down the real (trigonometric) Fourier series of F (t).
(b) Let g(t) = t for 0 < t < 2. Let G(t) be an odd periodic extension of g with period 4.
i. Sketch the graph of G(t) (show at least three periods).
ii. Write down the complex Fourier series of G(t).
iii. Write down the real (trigonometric) Fourier series of G(t).
π
2. For L > 0, let ωk = k L
, for k = 0, ±1, ±2, ±3, ±4, · · · , and let ∆ω =
π
L.
(a) Show that the Fourier series of a 2L-periodic function f (t) can be written as
f (t) =
∞
1 X e
f (ωk )eiωk t ∆ω,
2π
k=−∞
where i =
√
−1 and
Z
L
fe(ωk ) =
f (t)e−iωk t dt.
−L
(b) Use the notation of (a). Assume L > 1. Let g(t) be the 2L-periodic function given by


1 if − 1/2 < t < 1/2,
g(t) = 0 if − L < t ≤ −1/2,


0 if 1/2 ≤ t < L.
Compute ge(ωk ) =
RL
−L
g(t)e−iωk t dt, for all integer k.
(c) For the function g(t) in (b), do the following.
i. Let L = 1. Sketch the graph of ge(ωk ) for −6π < ωk < 6π. For the values ωk , use the ω-axis (i.e.
the horizontal axis with ω variable)
ii. Let L = 2. Sketch the graph of ge(ωk ) for −6π < ωk < 6π. Use the ω-axis for the values ωk .
iii. Let L = 4. Sketch the graph of ge(ωk ) for −6π < ωk < 6π. Use the ω-axis for the values ωk .
3. Sketch the graph of the following functions and calculate their Fourier transforms. (For a given function
R∞
f (t) its Fourier transform is given by fb(ω) = −∞ f (t)e−iωt dt.)
(
(a) f (t) =
1 if 0 < t < 1,
0 otherwise.


if 0 < t ≤ 1,
1
(b) g(t) = −2 if 1 < t ≤ 2,


0
otherwise.
(
e−2t if 0 < t ≤ 1,
(c) h(t) =
0
otherwise.
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4. For real variable t, consider a complex-valued function z(t). For example, z(t) = e−t+2i . We say “z(t) → 0
as t → ∞” if the modulus (in other words, the magnitude) |z(t)| → 0 as t → ∞. Also, recall that by
R∞
RT
R∞
RT
definition −∞ f (t)dt = limT →∞ −T f (t)dt. Similarly, 0 f (t)dt = limT →∞ 0 f (t)dt.
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(a) Let z(t) = e−t+it . (Here i =
√
−1.) Show that z(t) → 0 as t → ∞. (Hint: first, compute |z(t)|.)
(b) Let
(
e−at
f (t) =
0
if t > 0,
otherwise.
where the constant a > 0. Compute the Fourier transform fb(ω), i.e.
RT
f (t)e−iωt dt and let T → ∞.)
−T
R∞
−∞
f (t)e−iωt dt. (Compute
(c) Let g(t) = e−a|t| where the constant a > 0. Compute the Fourier transform gb(ω), i.e.
RT
(Compute −T g(t)e−iωt dt and let T → ∞.)
R∞
−∞
g(t)e−iωt dt.
5. Find the Fourier transform of the following functions:
(
sin(2t) if − 1 < t < 1,
(Hint: express sin(2t) in terms of complex exponentials.)
(a) f (t) =
0
otherwise.
(b) g(t) = e−5|t| cos(2t). (Hint: express cos(2t) in terms of complex exponentials.)
*Staple your HW. From now on, you will get F IV E marks OFF if you do not staple your HW!
Note that the instructor will NOT provide stapler.
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