Legendre Polynomials

The solution to Legendre Differential Equations

$$R(t)=A_n{t}^n+B_n{t}^{-n-1}$$

Integrating the product of 2 Legendre polynomials yields the following:

$$\underset{-1}{\overset{1}{\int }}{P}_n(x)P_m(x)dx=\frac{2}{2n+1}{\delta }_{mn}$$ $${\delta }_{mn}\{\begin{array}{ll}0& m\ne n\\ 1& m=n\end{array}$$ $$F(x)=\sum _{n=p}^{\infty }{A}_n{P}_n(x)|P_m(x)=$$ $$\underset{-1}{\overset{1}{\int }}F(x)P_m(x)dx=\sum _{n=0}^{\infty }{A}_n\frac{2}{2n+1}{\delta }_{mn}=A_m\frac{2}{2m+1}$$ $$A_m=\frac{2m+1}{2}\underset{-1}{\overset{1}{\int }}F(x)P_m(x)dx$$