This is what I did: Take the function sqrt(z^2-1). I try to think of it as rotating the values of the contour on this function. I already know the resulting integral on top of the segment is -i*integral from -1 to 1. Substitution of z = i*x we get the contour for sqrt(z^2+1) witht the branch points. the problem is that sqrt (z^2-1) becomes isqrt(1-z^2) in absolute value thats where the i comes from in the original integral. Thus, we have the integral is really sqrt(1-z^2)now that its rotated we take the phases from each side and get negative 2I where we have subsituted in the original integral z = xe^ipi/2 to get the desired limits. Is this the right way to interpret the integral? because I dont know any other way to yield the right answer, but I dont know if this is the right thing to do in general. Please help, any help will be appreciated!