Loading [MathJax]/extensions/TeX/boldsymbol.js

 

 

 

Orthogonal polynomials, Legendre

We can proceed in a similar fashion in order to determine the coefficients of L_2 \begin{equation*} L_2(x) = a+bx+cx^2, \end{equation*} using the orthogonality relations \begin{equation*} \int_{-1}^1L_0(x)L_2(x)dx=0, \end{equation*} and \begin{equation*} \int_{-1}^1L_1(x)L_2(x)dx=0, \end{equation*} and the condition L_2(1)=1 we would get \begin{equation} L_2(x) = \frac{1}{2}\left(3x^2-1\right). \tag{12} \end{equation}