8. The first three Legendre polynomials are P0(x) = 1, P1(x) = x, and P2(x) = 1/2(3x2- 1). If x = cosθ , then P0( cosθ ) = 1 and P1( cosθ ) = cos θ . Show that P2( cosθ ) = 1/4( 3cos2θ + 1 ).
9. Use the results of problem 8, to find a Fourier-Legendre expansion ( F (θ) = )of F( θ ) = 1 - cos2θ .
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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Using the identities:
We see that:
Substituting it back in ...
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.