Compute a Fourier Series
Exercise. We warm up with a reminder of how one computes the Fourier
series of a given periodic function using the integral Fourier coefficient for­
mulas.
Compute the Fourier series for the period 2π continuous sawtooth func­
tion f (t) = |t| for −π ≤ t ≤ π.
Answer.
π
−2π −π
2π
Figure 1. Graph of the period 2π continuous sawtooth function.
The period is 2π, so the half-period L = π. Since f (t) = |t| for −π ≤
t ≤ π, it is an even function we know the Fourier sine coefficents bn must
be zero.
Computing the cosine coefficients we get: For n �= 0:
2 π
t cos(nt) dt
π 0
−π
�
�
�
−
n24π
2 t sin(nt)
cos(nt)
��π
2
n
=
+
=
((−
1
)
−
1
)
=
π
n
n2 �
0
n2 π
0
an =
1
π
� π
�
|t| cos(nt) dt =
For n = 0:
�
�
2 π
1 π
a0 =
|t| dt =
t dt = π.
π −π
π 0
Thus, f (t) has Fourier series
�
�
π
4
cos(3t) cos(5t)
f (t) =
−
cos t +
+
+···
2
π
32
52
=
π
4
−
2
π
cos(nt)
n2
n odd
∑
for n odd
for
even
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18.03SC Differential Equations��
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