Given a multifunction f on an open subset U of ℂ, a holomorphic branch of f is a holomorphic function on U which selects from the possible choices of f(z) for each z ∈ U. Essentially, the branch is a holomorphic choice of principal values for f. For example, every z ∈ ℂ which is not a positive real can be written uniquely as z = re^iθ‎ on a cut plane where r>0 and 0 < θ‎ < 2π‎. A holomorphic branch for the complex logarithm on this cut plane is then logr + iθ‎. Note that 0 is a branch point and that there is a discontinuity across the positive real axis of 2π‎i.