Branch Point - Algebraic Branch Points

Algebraic Branch Points

Let Ω be a connected open set in the complex plane C and ƒ:Ω → C a holomorphic function. If ƒ is not constant, then the set of the critical points of ƒ, that is, the zeros of the derivative ƒ'(z), has no limit point in Ω. So each critical point z₀ of ƒ lies at the center of a disc B(z₀,r) containing no other critical point of ƒ in its closure.

Let γ be the boundary of B(z₀,r), taken with its positive orientation. The winding number of ƒ(γ) with respect to the point ƒ(z₀) is a positive integer called the ramification index of z₀. If the ramification index is greater than 1, then z₀ is called a ramification point of ƒ, and the corresponding critical value ƒ(z₀) is called an (algebraic) branch point. Equivalently, z₀ is a ramification point if there exists a holomorphic function φ defined in a neighborhood of z₀ such that ƒ(z) = φ(z)(z − z₀)k for some positive integer k > 1.

Typically, one is not interested in ƒ itself, but in its inverse function. However, the inverse of a holomorphic function in the neighborhood of a ramification point does not properly exist, and so one is forced to define it in a multiple-valued sense as a global analytic function. It is common to abuse language and refer to a branch point w₀ = ƒ(z₀) of ƒ as a branch point of the global analytic function ƒ−1. More general definitions of branch points are possible for other kinds of multiple-valued global analytic functions, such as those that are defined implicitly. A unifying framework for dealing with such examples is supplied in the language of Riemann surfaces below. In particular, in this more general picture, poles of order greater than 1 can also be considered ramification points.

In terms of the inverse global analytic function ƒ−1, branch points are those points around which there is nontrivial monodromy. For example, the function ƒ(z) = z2 has a ramification point at z₀ = 0. The inverse function is the square root ƒ−1(w) = w1/2, which has a branch point at w₀ = 0. Indeed, going around the closed loop w = eiθ, one starts at θ = 0 and ei0/2 = 1. But after going around the loop to θ = 2π, one has e2πi/2 = −1. Thus there is monodromy around this loop enclosing the origin.