The first three Legendre polynomials are P0(x) 1, P1(x) x, and P2(x) 1 2(3x 2 1). If x
Chapter 12, Problem 17(choose chapter or problem)
The first three Legendre polynomials are \(P_{0}(x)=1\), \(P_{1}(x)=x\), and \(P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)\). If \(x=\cos \theta\), then \(P_{0}(\cos \theta)=1\) and \(P_{1}(\cos \theta)=\cos \theta\). Show that \(P_{2}(\cos \theta)=\frac{1}{4}(3 \cos 2 \theta+1)\).
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