1. The problem statement, all variables and given/known data I am reading the solution the integral of (log(z))^2/(1+z^2) from 0 to infinity in a textbook, and I'm not sure I quite understand it, and I think this misunderstanding stems from my difficulty w/ branch points/cuts for multivalued functions. 2. Relevant equations 3. The attempt at a solution I've attached the solution. They appear to have drawn a contour that is a large semicircle in the UHP, avoiding the branch point at z=0 and have chosen a branch cut along the negative real axis. I have a couple questions here: (1) The choice of branch cut seems to be by convenience, but I don't understand the logic behind the choice of the negative real axis. (2) Why in the integral from infinity to 0 (see line above line 13.1) do they add i*pi? This makes somewhat sense to me as at the beginning (integral from 0 to infinity), you have r*e^(i*0), whose ln is ln(r), and at the line (infinity to 0) you have r*e^(i*pi), whose ln is ln(r)+i*pi... But I thought that the value didn't change until you crossed the branch point? At least, that is the way it would be if you had drawn "keyhole" contour. Any help?