Legendre Polynomial Orthogonality and Size

Question

Show $(P_i,P_j)=\begin{cases} 0& i \neq j \\ \frac{2}{2j+1} & i = j\end{cases}$ for $0 \leq i, j\leq2$

I'm just not sure exactly what I'm supposed to do. Do I plug in values of i and j and solve?

Answer 1

You have to show the orthogonality relations of the first three Legendre polynomials (not Lagrange): $$P_0(x)=1,\quad P_1(x)=x,\quad P_2(x)=\frac{1}{2}(3x^2-1),$$ where $(.,.)$ is defined as $$(P_i,P_j) = \int_{-1}^{1}P_i(x)P_j(x) dx \,.$$ Here are two examples $$(P_0,P_1) = \int_{-1}^{1}P_0(x)P_1(x)= \int_{-1}^{1}1\cdot xdx = \Big[\frac{x^2}{2}\Big]_{-1}^{1} = 0 $$ $$(P_1,P_1) = \int_{-1}^{1}x\cdot xdx = \Big[\frac{x^3}{3}\Big]_{-1}^{1} = \frac{1}{3}-\frac{-1}{3}=\frac{2}{3} =\frac{2}{2\times 1 +1} $$ I guess you can now compute the remaining $(P_i,P_j).$

Answer 2

Best directly solve it for all $i,j$. Orthogonality of $P_i$ and $P_j$ when $i \not= j$ is easy; for instance use the fact that the Legendre equation is a special form of Sturm-Liouville equations.

For the second part, see my post on (Legendre Polynomials: proofs).