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Math 401 (Sec. 502)
Spring, 2009
Homework 8 (due March 26)
Q1. (a)
Show that any function
f
(
x
) deﬁned on [
−
L, L
] can be written as
the sum of an even function and an odd function.
[
Hint.
Suppose
f
(
x
) =
p
(
x
) +
q
(
x
), where
p
(
x
) is even and
q
(
x
) is odd.
Then solve the equations,
f
(
x
) =
p
(
x
) +
q
(
x
) and
f
(
−
x
) =
p
(
x
)
−
q
(
x
),
simultaneously for
p
(
x
) and
q
(
x
) in terms of
f
(
x
) and
f
(
−
x
).]
(b)
Assuming that
f
′
(
x
) is deﬁned on [
−
L, L
], prove that,
f
′
(
x
) is even
when
f
(
x
) is odd and
f
′
(
x
) is odd when
f
(
x
) is even.
Q2.
If
f
(
x
) is continuous on the interval 0
< x < a
, is its even periodic
extension continuous? What about the odd periodic extension? [
Hint.
check
at
x
= 0 and
±
a
.]
Q3.
Sketch both the even and odd extensions of the functions
(a)
f
(
x
) = 1
,
0
< x <
2
(b)
f
(
x
) =
x,
0
< x <
1
(c)
f
(
x
) = cos
x,
0
< x < π
.
Q4.
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 Spring '09
 SARKAR
 Math

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