Define a branch of the function

$\displaystyle f(z) = ( {z-1 \over z+1})^ {1 \over 3}$

that is regular everywhere except for a cut along the real interval [-1,1] and that takes real positive values for real z>1.

I understand that a branch cut restricts a multi-valued function to a single-valued function, but won't there be several branches here?