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Section 6.2 SturmLiouville Theory
95
For the third integral, we have
±
1

1
g
(
x
)
h
(
x
)
w
(
x
)
dx
=
±
1

1
2
x
(

1+4
x
2
)
²
1

x
2
dx
=0
,
because we are integrating an odd function over a symmetric interval.
9.
In order for the functions 1 and
a
+
bx
+
x
2
to be orthogonal, we must have
±
1

1
1
·
(
a
+
bx
+
x
2
)
dx
Evaluating the integral, we ±nd
ax
+
b
2
x
2
+
1
3
x
3
³
³
³
1

1
=2
a
+
2
3
a
=

1
3
.
In order for the functions x and
1
3
+
bx
+
x
2
to be orthogonal, we must have
±
1

1
1
·
(
1
3
+
bx
+
x
2
)
xdx
Evaluating the integral, we ±nd
1
6
x
2
+
b
3
x
3
+
1
4
x
4
³
³
³
1

1
=
b
3
b
.
13.
Using Theorem 1, Section 5.6, we ±nd the norm of
P
n
(
x
)tobe
±
P
n
±
=
´±
1

1
P
n
(
x
)
2
dx
µ
1
2
=
´
2
2
n
+1
µ
1
2
=
√
2
√
2
n
.
Thus the orthonormal set of functions obtained from the Legendre polynomials is
√
2
√
2
n
P
n
(
x
)
,n
,
2
,....
17.
For Legendre series expansions, the inner product is de±ned in terms of
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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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