SOLUTION:
Given That
f(x)=sin(2πx) with
m=1,2,.......M
So
The Legendre polynomials form
a complete Orthogonal system over the interval
[-1,1]
with respect to the weight function==> ,
The function may be expanded
interms of them as
To obtain, the coefficient in the expansion
multiply both sides by and
integrate,
where the coefficient
Here
To find the coefficients :
The Legendre's Polynomial is as follows:
To find
To find
To find
To find
To find
To find
From the above calculations,
we observe that
,
That is , all even vanish. is an
odd function.
,
,
WI) = 1
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(1) - Σα, ΡΑ(α)
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Pm (2)
1 pcm)f(x)dır Əa | Pa(w)Pm(a)dr n=0 J-1
an = + | Pn(z)f(x)dx J-1
f(1) = sin(272)
do, d1, 02, ...
Pol.2) = 1
P1(x) = 2
P2(x) = 5(3.- 1)
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P5(x) == (6325 – 70x3 + 152)
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n = 5= P3(x) = (63x5 – 70x} + 152)
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