Hints from Bell about Legendre Polynomials
See p. 180 for a list of the first few Legendre Polynomials.
The even numbered Legendre Polynomials only involve even powers of x, so they are even functions.
The odd numbered Legendre Polynomials only involve odd powers of x, so they are odd functions.
The Legendre Polynomials are orthogonal on the interval [-1,1].
p. 209, 5. asks you to show that
$ P_n(\cos\theta) $
are orthogonal on [0,pi] with respect to the weight function
$ \sin\theta, $
i.e., to show that
$ \int_0^\pi P_n(\cos\theta)P_m(\cos\theta)\sin\theta\ d\theta=0 \qquad\text{if }n\ne m. $
The key here is to use the change of variables
$ x=\cos\theta $
and convert the integral to one in x over the interval [-1,1], where you can use the orthogonality of the Legendre Polynomials. (You don't have to show that the Legendre Polynomials are orthogonal on [-1,1]. That's a given.)
p. 216, problems 1 and 3 ask you to expand a given function in terms of Legendre Polynomials. Here, you will use the fact that if Q(x) is a polynomial of degree N, then
$ Q(x)=\sum_{n=0}^N c_nP_n(x) $
where the coefficients c_n are computed via orthogonality:
$ \int_{-1}^1 Q(x)P_m(x)\ dx=c_m\int_{-1}^1 P_m(x)P_m(x)\ dx. $
You will need to use the fact given on page 212 that
$ \int_{-1}^1 P_m(x)^2\ dx=\frac{2}{2m+1}. $
p. 216, 5. asks you to show that if f(x) is even, then all the odd coefficients in its Legendre expansion must vanish, i.e., that
$ \int_{-1}^1 f(x)P_n(x)\ dx=0 $
if n is odd. Recall that if n is odd, P_n is odd. An even times an odd is a ... etc.
