As in Problem 29 , find the first three coefficients in the expansion f ( x ) = c 0 P 0 ( x ) + c 1 P 1 ( x ) + c 2 P 2 ( x ) + … , When f ( x ) = | x | , − 1 < x < 1 In Section 8.8 , it was shown that the Legendre polynomials P n ( x ) are orthogonal on the interval [ − 1 , 1 ] with respect to the weight function w ( x ) ≡ 1 . Using the fact that the first three Legendre polynomials are P 0 ( x ) ≡ 1 , P 1 ( x ) = x , P 2 ( x ) = ( 3 2 ) x 2 − ( 1 2 ) , find the first three coefficients in the expansion f ( x ) = c 0 P 0 ( x ) + c 1 P 1 ( x ) + c 2 P 2 ( x ) + … , where f ( x ) is the function f ( x ) : = { − 1 , − 1 < x < 0 , 1 , 0 < x < 1 .
As in Problem 29 , find the first three coefficients in the expansion f ( x ) = c 0 P 0 ( x ) + c 1 P 1 ( x ) + c 2 P 2 ( x ) + … , When f ( x ) = | x | , − 1 < x < 1 In Section 8.8 , it was shown that the Legendre polynomials P n ( x ) are orthogonal on the interval [ − 1 , 1 ] with respect to the weight function w ( x ) ≡ 1 . Using the fact that the first three Legendre polynomials are P 0 ( x ) ≡ 1 , P 1 ( x ) = x , P 2 ( x ) = ( 3 2 ) x 2 − ( 1 2 ) , find the first three coefficients in the expansion f ( x ) = c 0 P 0 ( x ) + c 1 P 1 ( x ) + c 2 P 2 ( x ) + … , where f ( x ) is the function f ( x ) : = { − 1 , − 1 < x < 0 , 1 , 0 < x < 1 .
Solution Summary: The author calculates the coefficients in the given expansion by the formula, m=0,1,2,dots
As in Problem 29, find the first three coefficients in the expansion
f
(
x
)
=
c
0
P
0
(
x
)
+
c
1
P
1
(
x
)
+
c
2
P
2
(
x
)
+
…
,
When
f
(
x
)
=
|
x
|
,
−
1
<
x
<
1
In Section 8.8, it was shown that the Legendre polynomials
P
n
(
x
)
are orthogonal on the interval
[
−
1
,
1
]
with respect to the weight function
w
(
x
)
≡
1
. Using the fact that the first three Legendre polynomials are
P
0
(
x
)
≡
1
,
P
1
(
x
)
=
x
,
P
2
(
x
)
=
(
3
2
)
x
2
−
(
1
2
)
,
find the first three coefficients in the expansion
f
(
x
)
=
c
0
P
0
(
x
)
+
c
1
P
1
(
x
)
+
c
2
P
2
(
x
)
+
…
,
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