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Section 4.8
Bessel Series Expansions
75
13.
Use (6) with
p
= 4. Then
J
5
(
x
)=
8
x
J
4
(
x
)

J
3
(
x
)
=
8
x
±
6
x
J
3
(
x
)

J
2
(
x
)
²

J
3
(
x
) (by (6) with
p
=3)
=
³
48
x
2

1
´
J
3
(
x
)

8
x
J
2
(
x
)
=
³
48
x
2

1
´³
4
x
J
2
(
x
)

J
1
(
x
)
´

8
x
J
2
(
x
)
(by (6) with
p
=2)
=
³
192
x
3

12
x
´
J
2
(
x
)

³
48
x
2

1
´
J
1
(
x
)
=
12
x
³
16
x
2

1
´±
2
x
J
1
(
x
)

J
0
(
x
)
²

³
48
x
2

1
´
J
1
(
x
)
(by (6) with
p
=1)
=

12
x
³
16
x
2

1
´
J
0
(
x
)+
³
384
x
4

72
x
2
+1
´
J
1
(
x
)
.
17.
(a) From (17),
A
j
=
2
J
1
(
α
j
)
2
µ
1
0
f
(
x
)
J
0
(
α
j
x
)
xdx
=
2
J
1
(
α
j
)
2
µ
c
0
J
0
(
α
j
x
)
=
2
α
2
j
J
1
(
α
j
)
2
µ
cα
j
0
J
0
(
s
)
sds
(let
α
j
x
=
s
)
=
2
α
2
j
J
1
(
α
j
)
2
J
1
(
s
)
s
¶
¶
¶
¶
¶
cα
j
0
=
2
cJ
1
(
α
j
)
α
j
J
1
(
α
j
)
2
.
Thus, for 0
<x<
1,
f
(
x
∞
·
j
=1
2
cJ
1
(
α
j
)
α
j
J
1
(
α
j
)
2
J
0
(
α
j
x
)
.
(b) The function
f
is piecewise smooth, so by Theorem 2 the series in (a) converges
to
f
(
x
) for all 0
1, except at
x
=
c
, where the series converges to the average
value
f
(
c
+)+
f
(
c

)
2
=
1
2
.
21.
(a) Take
m
=1
/
2 in the series expansion of Exercise 20 and you’ll get
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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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