MATH 54 − QUIZ 11 − SOLUTIONS
PEYAM RYAN TABRIZIAN
1. (10 points) Find the (full) Fourier series of f (x) = |x| on (−π, π)
All we need to do is to find Am and Bm such that:
|x| “ = ”
∞
X
Am cos(mx) + Bm sin(mx)
m=0
Because |x| is even on (−π, π) and |x| = x on (0, π), we get:
Rπ
A0 =
=
R−ππ
2
|x| dx
R−π
π
1dx
|x| dx
Z 2π
0
1 π
xdx
π 0
π 2 /2
=
π
π
=
2
=
And if m ≥ 1, integrating by parts, we get:
Date: Thursday, April 30, 2015.
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PEYAM RYAN TABRIZIAN
Rπ
Am =
R−ππ
R−π
π
cos2 (mx)dx
x cos(mx)dx
π Z π
π sin(mx)
sin(mx)
2
x
dx
=
−
π
m
m
0
0
π 2
π sin(πm) 0 sin(0)
− cos(mx)
=
−
−
π
m
m
m2
0
2
cos(πm) cos(0)
=
0−0+
−
π
m2
m2
(−1)m − 1
2
=
π
m2
m
2((−1) − 1)
=
π (m2 )
And B0 = 0 by convention, and Bm = 0 if m ≥ 1, because |x| is
even on (−π, π).
=
2
|x| cos(mx)dx
0
Therefore:
∞
π X
|x| “ = ” +
2 m=1
2((−1)m − 1)
π (m2 )
cos(mx)