I'm given two solutions to Legendre's equation:
$P_1=x$
$Q_0=\frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)$
I'm trying to explain why their overlap integral (i.e. $\int_{-1}^{1} P_1 Q_0 dx$) is non-zero. I computed it and it is indeed non-zero, but I'm having a difficult time justifying why that is. I'm thinking it has something to do with that fact that the $P_n$ and $Q_n$ solutions are constructed w.r.t different weight functions. Or perhaps it has something to do with the completeness of solutions. Any thoughts?