Branch point

In the mathematical field of complex analysis, a branch point of a multivalued function is a point such that if the function is ${\displaystyle n}$-valued (has ${\displaystyle n}$ values) at that point, all of its neighborhoods contain a point that has more than ${\displaystyle n}$ values.^[1] 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 ${\displaystyle w^{2}=z}$ for ${\displaystyle w}$ as a function of ${\displaystyle z}$. 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 ${\displaystyle w}$ 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.^[2] In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.