Given the following defining equation for Legendre polynomials M P (x) = _(-1) m - (2n -2m)! yo-2m . M= n /2 n-even m=0 2" (n-m)!(n-2m)! (n-1)/2 n- odd a. Generate the polynomials for n = 0, 1, 2. The coefficients for the Fourier-Legendre series, f(x)- EanPr(x) n=0 2n + 1 1 are given by an= 2 -Sf ( x) Pn(x) dx -1 b. Calculate the coefficients (for n = 0, 1 and 2) of the Fourier-Legendre series for the function f(x) = x. c. Use Rodrigues's formula, Pr(x) = - 1 an 2" n! dx" [(x2 -1)"] to generate the same Legendre polynomials as in part a. (that is for n = 0, 1, 2). Show all work! d. I Given an even function, f(x) show that the odd coefficients generated by an 2n+ 1 " f(x) P. (x)dx are zero and alternately for an odd function 2 -1 the even coefficients are zero. [Make sure to do this for any even or odd function f(x) depending on the case being derived.]