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6
Chapter 2
Fourier Series
Solutions to Exercises 2.2
1.
The graph of the Fourier series is identical to the graph of the function, except
at the points of discontinuity where the Fourier series is equal to the average of the
function at these points, which is
1
2
.
5.
We compute the Fourier coeﬃcients using he Euler formulas. Let us ±rst note
that since
f
(
x
)=

x

is an even function on the interval

π<x<π
, the product
f
(
x
) sin
nx
is an odd function. So
b
n
=
1
π
±
π

π
odd function
²
³´
µ

x

sin
nx dx
=0
,
because the integral of an odd function over a symmetric interval is 0. For the other
coeﬃcients, we have
a
0
=
1
2
π
±
π

π
f
(
x
)
dx
=
1
2
π
±
π

π

x

dx
==
1
2
π
±
0

π
(

x
)
dx
+
1
2
π
±
π
0
xdx
=
1
π
±
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.
 Fall '11
 StuartChalk

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