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Differential Equations by Pearson
book

Differential Equations by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
472 pages
13h 25m
English
Pearson India
Content preview from Differential Equations by Pearson
Integration in Series 6-63
If n is even, the last term is
0
2
1
.().
2
n
n
cTx
Integration of T
n
(x)
Consider
[]
11
( ) cos (cos ) cos sin
1
sin( 1) sin( 1)
2
cos( 1) cos( 1)
1
1
21 1
11 1
() ()
21 1
n
nn
Txdx n d n d
nnd
nn
n
nn
Tx Tx
nn
θθ θθθ
θθθ
θθ
+−
=⋅ =−
=− +
+−
⎡⎤
=− >
⎢⎥
+−
⎣⎦
⎡⎤
=⋅−⋅
⎢⎥
+−
⎣⎦
∫∫
The above result does not hold good for
0n =
and 1. For these
values of n we have
01
22
2
1
02
() 1 ()
cos 1 1
( ) (2cos ) (1 cos 2 )
224 4
1
().
4
T x dx dx x T x
x
T x dx x dx
TT
θ
θθ
=⋅==
=== = =+
=+
∫∫
∫∫
Orthogonality of Chebyshev Polynomials T
n
(x)
Prove that
1
2
1
0, if
() ()
,if 0
1
,if 0
2
mn
mn
TxTx
dx m n
x
mn
π
π
===
=≠
Solution Case (i) 0mn==. In this case, we get (since
0
1T =
)
1
1
111
1
2
1
1
sin sin 1
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Publisher Resources

ISBN: 9781299487406Publisher Website