I was asked to use antiderivative to evaluate the integral of z^(1/2) dz , over contour C1. for which the intergrand is the branch z^(1/2) = r^(1/2)exp(ix/2) , (r>0, 0<x<2*pi) and C1 is the contour from z=-3 to z=3. The books says the integrand is piecewise continuous on C1 and integral exist there but the branch of z^(1/2) is not defined on the ray x=0 (or pt z=3). And so we have to integrate it over another branch f(z) = r^(1/2)exp(ix/2) , (r>0, -pi/2<x<3*pi/2) I don't really understand why the branch is not defined at x=0. and also, why do we choose -pi/2 to 3pi/2 as our new range for the branch? How do we determine that? thanks a lot!