The two branches are given by g_0(z)= z+ (z^2- 1)^{1/2} and g_1(z)= z- (z^2- 1)^{1/2}, taking the + and - values of the square root.

1/g_1(z)= 1/(z- (z^2- 1)^{1/2}). Rationalize the denominator by multiplying numerator and denominator by z+ (z^2- 1)^{1/2}.