Swinburne University of Technology
School of Science, Computing and Engineering Technologies
MTH20011
Tutorial 2
Topic: Fourier series and Fourier coefficients
For the following functions, sketch three periods of the function and find the Fourier
series. Use the following identities to calculate the Fourier coefficients
Z
t sin(at) cos(at)
+ C, (a 6= 0)
+
a
a2
Z
t cos(at) sin(at)
+ C, (a 6= 0)
t sin(at) dt = −
+
a
a2
Z
cos((a + b)t) cos((a − b)t)
sin(at) cos(bt) dt = −
−
+ C, (a 6= b)
2(a + b)
2(a − b)
Z
sin((a + b)t) sin((a − b)t)
+
+ C, (a 6= b)
sin(at) sin(bt) dt = −
2(a + b)
2(a − b)
t cos(at) dt =
1.
2.
3.

−π ≤ t < 0
 0
t
0≤t<π
f (t) =

f (t + 2π) for all t.

−1 ≤ t < 0
 −t
t
0≤t<1
x(t) =

x(t + 2) for all t.

0≤t<1
 1−t
−1 + t 1 ≤ t < 2
y(t) =

y(t + 2) for all t.
4.
f (t) =
sin(πt) 0 ≤ t < 1
f (t + 1) for all t.
5.
f (t) = 2 sin(t) + sin(2t), for all t.
Mathematics 4A
Page 1 of 2
Swinburne University of Technology
School of Science, Computing and Engineering Technologies
MTH20011
Tutorial 2
Answers
1.
∞ π X
f (t) ∼ +
4
n=1
(−1)n − 1
(−1)n
cos(nt)
−
sin(nt)
πn2
n
2.
∞
1 4 X cos((2n + 1)πt)
1 4
x(t) ∼ − 2
= − 2
2
2 π
(2n + 1)
2 π
n=0
1
1
cos(πt) + cos(3πt) +
cos(5πt) + . . .
9
25
3.
∞
1 4 X cos((2n + 1)πt)
1 4
y(t) = 1−x(t) ∼ + 2
= + 2
2
2 π
(2n + 1)
2 π
n=0
4.
1
1
cos(πt) + cos(3πt) +
cos(5πt) + . . .
9
25
∞
f (t) ∼
4 X cos(2πnt)
2
−
π π
4n2 − 1
n=1
5.
f (t) = 2 sin(t) + sin(2t).
Mathematics 4A
Page 2 of 2