Editing Branch point (section)

{{Short description|Point of interest for complex multi-valued functions}}

In the [[mathematics|mathematical]] field of [[complex analysis]], a '''branch point''' of a [[multivalued function]] is a point such that if the function is <math>n</math>-valued (has <math>n</math> values) at that point, all of its neighborhoods contain a point that has more than <math>n</math> values.<ref>{{Citation |last=Das |first=Shantanu |title=Fractional Differintegrations Insight Concepts |date=2011 |url=http://dx.doi.org/10.1007/978-3-642-20545-3_5 |work=Functional Fractional Calculus |pages=213–269 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-20545-3_5 |isbn=978-3-642-20544-6 |access-date=2022-04-27}} (page 6)</ref><!-- not sure how to handle infinite order branch point (if those even exist). this could use an expert's second opinion. --> Multi-valued functions are rigorously studied using [[Riemann surface]]s, 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 <math>w^2=z</math> for <math>w</math> as a function of <math>z</math>.  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 <math>w</math> 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.<ref>{{harvnb|Ahlfors|1979}}</ref>  In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.