Legendre polynomials a) The Legendre polynomials P(r) are related to a "generating function" by ∞ 1 = ΣP(x)t¹ √1-2tx + t

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Legendre polynomials a) The Legendre polynomials P(r) are related to a "generating function" by ∞ 1 = ΣP(x)t¹ √1-2tx + t² 1=0 for t < 1 and r ≤ 1. Find an expression for P(x) by differentiating this equation / times with respect to t and setting t = 0 afterward. Use the result to show that Po(x) = 1, P₁(x) = x, P₂ = (3x²-1). b) Express the polynomials f(x) = 3x² +1 and f(x) = x² - 2x +4 in terms of P(x). c) Replacing the argument z with cos , express the trigonometric functions f(1) = sin² +3 and f(1) = 2 cos (20) in terms of Pi (cos).