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# Bessel and Legendre Series : Fourier-Legendre Expansions

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8. The first three Legendre polynomials are P0(x) = 1, P1(x) = x, and P2(x) = 1/2(3x2- 1). If x = cos&#952; , then P0( cos&#952; ) = 1 and P1( cos&#952; ) = cos &#952; . Show that P2( cos&#952; ) = 1/4( 3cos2&#952; + 1 ).

9. Use the results of problem 8, to find a Fourier-Legendre expansion ( F (&#952;) = )of F( &#952; ) = 1 - cos2&#952; .

10. Why is a Fourier-Legendre expansion of a polynomial function that is defined on the interval ( -1, 1 ) necessarily a finite series.

11. Using only your conclusions from problem 10, find the finite Fourier-Legendre series of f(x) = x2

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If

Then:

Using the identities:

We see that:

Substituting it back in ...

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Bessel and Legendre Series and Fourier-Legendre Expansions are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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