The Legendre function of second kind $Q_\nu(z)$ has branch points at $z=\pm 1$. The branch points are joined by a cut along the real axis. Show that $$Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$$ is single valued with the real axis $-1\leq x\leq 1$ taken as a cut line.
I'm quite able to see it, Why this is shown. But not able to write a formal proof for it. We can write the function $$Q_0(z)=\frac{1}{2}\left[\ln(z+1)-\ln(z-1)\right]=\frac{1}{2}[f(z)-g(z)]$$ Now the function $f(z)$ is single values expect a branch lying in real axis $x<-1$. Similarly $g(z)$ has a branch lying in real axis $x<1$. In the region of coincidence of branch, the effect cancel out due to negative sign $(f-g)$. Hence we have branch in between $-1$ and $1$.
Can anyone tell me how do I can show in writing?