Chapter 8.8, Problem 31E

To determine

(a)

To show:

The orthogonality condition for Legendre polynomials implies that ${\displaystyle {\int }_{-1}^1{P}_n\left(x\right)q\left(x\right)dx}=0$.

To determine

(b)

To prove:

If $Q_n\left(x\right)$ is a polynomial of degree n such that ${\displaystyle {\int }_{-1}^1{Q}_n\left(x\right)P_k\left(x\right)dx}=0$ for $k=0,1,\mathrm{...},n-1$, then $Q_n\left(x\right)=c{P}_n(x)$ for some constant c.