Branch point

In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation w2 = z for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an appropriate sense, is continuous at the origin. This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity. In geometric function theory, unqualified use of the term branch point typically means the former more restrictive kind: the algebraic branch points. In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type. Let be a connected open set in the complex plane and a holomorphic function. If is not constant, then the set of the critical points of , that is, the zeros of the derivative , has no limit point in . So each critical point of lies at the center of a disc containing no other critical point of in its closure. Let be the boundary of , taken with its positive orientation. The winding number of with respect to the point is a positive integer called the ramification index of .