Chapter 10.3, Problem 29E

In Section 8.8, it was shown that the Legendre polynomials $P_n\left(x\right)$ are orthogonal on the interval $\left[-1,1\right]$ with respect to the weight function $w\left(x\right)\equiv 1$. Using the fact that the first three Legendre polynomials are

$P_0\left(x\right)\equiv 1$, $P_1\left(x\right)=x$, $P_2\left(x\right)=\left(\frac{3}{2}\right)x^2-\left(\frac{1}{2}\right)$,

find the first three coefficients in the expansion

$f\left(x\right)=c_0{P}_0\left(x\right)+c_1{P}_1\left(x\right)+c_2{P}_2\left(x\right)+\dots$,

where $f\left(x\right)$ is the function

$f\left(x\right):=\{\begin{array}{l}-1,-1<x<0,\\ 1\text{, }0<x<1\end{array}$.