082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

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Example16.1 Show that f (x) =xandg(x) =x −1 areorthogonal on 0, 3 ]
.
2
16.2 Orthogonal functions 481
Solution
] 3/2
∫ 3/2 ∫ 3/2
[ x
x(x−1)dx= x 2 3
−xdx=
0
0 3 − x2
2
0
[
Hence f andgare orthogonal over the interval 0, 3 ]
.
2
= 9 8 − 9 8 = 0
Clearlyfunctionsmaybeorthogonaloveroneintervalbutnotorthogonaloverothers.
For example,
∫ 1
0
x(x−1)dx≠0
and soxandx −1 arenotorthogonal over [0,1].
Example16.2 Show f (t) = 1,g(t) = sint andh(t) = cost aremutually orthogonal over [−π,π].
Solution We are required toshow thatany pairoffunctions isorthogonal over [−π,π].
∫ π
−π
1sintdt =[−cost] π −π = −cosπ +cos(−π)
∫ π
−π
= −(−1) + (−1) =0
1costdt =[sint] π −π = sinπ −sin(−π) =0
Usingthe trigonometricidentitysin2A = 2sinAcosA, wecan write
∫ π
−π
sintcostdt =
∫ π
−π
[
1 cos(2t)
2 sin(2t)dt = − 4
cos(2π) −cos(−2π)
= − = 0
4
] π
−π
Hence the functions 1,sint, cost form an orthogonal set over [−π,π].
ThesetofExample 16.2 maybeextendedto
{1,sint,cost,sin(2t),cos(2t),sin(3t),cos(3t),...,sin(nt),cos(nt)}
n ∈ N
Example16.3 Verifythat {1,sint,cost,sin(2t),cos(2t),...} formsanorthogonal setover [−π,π].
Solution Supposen,m ∈ N. We must show that all combinations of 1,sinnt, sinmt, cosnt and
cosmt are orthogonal.
∫ π
[ −cos(nt) −cos(nπ) +cos(−nπ)
1sin(nt)dt =
= = 0
n n
−π
] π
−π