Find the expansion of the polynomial f(x) = -3 - 2x - x^2 in terms of the Legendre polynomials. P(x) = 1, Pi = x, Px = 3 - x Note: Recall that the Legendre polynomials are orthogonal on the interval [-1,1]. f(x) = 0
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The Legendre polynomial P0(x) = 1, so we can write P0(x) = P(x). The Legendre polynomial P1(x) = x, so we can write P1(x) = Pi(x). The Legendre polynomial P2(x) can be expressed as P2(x) = Px(x) - (1/3)P(x). Plugging in the given values, we have P2(x) = (3 - x) Show more…
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In Section $8.8,$ it was shown that the Legendre polynomials $P_{n}(x)$ are orthogonal on the interval $[-1,1]$ with respect to the weight function $w(x) \equiv 1 .$ Using the fact that the first three Legendre polynomials are $$P_{0}(x) \equiv 1, P_{1}(x)=x, P_{2}(x)=(3 / 2) x^{2}-(1 / 2)$$ find the first three coeffcients in the expansion $$f(x)=c_{0} P_{0}(x)+c_{1} P_{1}(x)+c_{2} P_{2}(x)+\cdots$$ where $$f(x)$$ is the function $$f(x) : \left\{\begin{array}{cc}{-1,} & {-1< x < 0} \\ {1,} & {0 < x < 1}\end{array}\right.$$
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