Is this question from the exercises of chapter 5 of the text 'Spectral Methods' by Trefethen? If so, I believe equation (5.9) of that text is wrong. The potential should have the form
phi(z) = log(|z+sqrt(z^2 - 1)/2|)
so that the minus sign is actually a plus. I'm working on the first exercise. Your question seems to be pertaining to the fifth exercise. Not sure if this helps.