Question

3-11. Expand the generating function (1-2μχ + μ2)-1/2 in the form of Eq. (3.47), to show that the first few coefficients are the Legendre polynomials given in Eqs. (3.41).

Answer 1

F (x, m) = (1 - 2xmu + mu^2)^-1/2 = [1 + (- 2xmu + mu^2)]^-1/2 F (x, mu) = 1 - 1/2 (2xmu + mu) + (-1/2) (-1/2 - 1)/2 + (-2xn + m^2)^2 + (1/2)(-1/2 - 1) (-1/2 -2)/6 (-2mu + mu^2)^3 +...= 1 + xmu - mu^2/2 + 3/8 (4x^2 mu^2 + m^2 - 4 x mu^3 0 - 3/6 times 4 times 5/2 [- x^3 mu^3 + mu^6 - 3xmu^3/- 2xmu + mu^2] +.. = 1 + x mu - mu^2/2 + 3/2 x^2 mu^2 + 3/6 mu^4 - 3/2 xmu^3 - 15/4 phi times - phi x^3 m^3 - 15/4 phi mu^6 - 15/4 phi x 6 x^2 mu^4 + 15/40 3 x mu^4 +..= 1 + mu x + mu^2 (-1/2 + 3/2 x^2) + mu^2 (- 3/2 x + 15 x^2/6) +.. & given F (x, mu) = sigma^infinity_l = 0 mu^p_l (x) = p_0 (x) + mu p_1 (x) + mu^2 p_2 (x) + mu^5 p-3 (x) +..comparing eqn 1 & 2 p_0 (x) = 1 & p_1 (x) = x & p_2 (x) = 3/2