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. 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. In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.

Let ${\displaystyle \Omega }$ be a connected open set in the complex plane ${\displaystyle \mathbb {C} }$ and ${\displaystyle f:\Omega \to \mathbb {C} }$ a holomorphic function. If ${\displaystyle f}$ is not constant, then the set of the critical points of ${\displaystyle f}$, that is, the zeros of the derivative ${\displaystyle f'(z)}$, has no limit point in ${\displaystyle \Omega }$. So each critical point ${\displaystyle z_{0}}$ of ${\displaystyle f}$ lies at the center of a disc ${\displaystyle B(z_{0},r)}$ containing no other critical point of ${\displaystyle f}$ in its closure.

Let ${\displaystyle \gamma }$ be the boundary of ${\displaystyle B(z_{0},r)}$, taken with its positive orientation. The winding number of ${\displaystyle f(\gamma )}$ with respect to the point ${\displaystyle f(z_{0})}$ is a positive integer called the ramification index of ${\displaystyle z_{0}}$. If the ramification index is greater than 1, then ${\displaystyle z_{0}}$ is called a ramification point of ${\displaystyle f}$, and the corresponding critical value ${\displaystyle f(z_{0})}$ is called an (algebraic) branch point. Equivalently, ${\displaystyle z_{0}}$ is a ramification point if there exists a holomorphic function ${\displaystyle \phi }$ defined in a neighborhood of ${\displaystyle z_{0}}$ such that ${\displaystyle f(z)=\phi (z)(z-z_{0})^{k}+f(z_{0})}$ for integer ${\displaystyle k>1}$.

Typically, one is not interested in ${\displaystyle f}$ 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 ${\displaystyle w_{0}=f(z_{0})}$ of ${\displaystyle f}$ as a branch point of the global analytic function ${\displaystyle f^{-1}}$. 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 ${\displaystyle f^{-1}}$, branch points are those points around which there is nontrivial monodromy. For example, the function ${\displaystyle f(z)=z^{2}}$ has a ramification point at ${\displaystyle z_{0}=0}$. The inverse function is the square root ${\displaystyle f^{-1}(w)=w^{1/2}}$, which has a branch point at ${\displaystyle w_{0}=0}$. Indeed, going around the closed loop ${\displaystyle w=e^{i\theta }}$, one starts at ${\displaystyle \theta =0}$ and ${\displaystyle e^{i0/2}=1}$. But after going around the loop to ${\displaystyle \theta =2\pi }$, one has ${\displaystyle e^{2\pi i/2}=-1}$. Thus there is monodromy around this loop enclosing the origin.

Suppose that g is a global analytic function defined on a punctured disc around z₀. Then g has a transcendental branch point if z₀ is an essential singularity of g such that analytic continuation of a function element once around some simple closed curve surrounding the point z₀ produces a different function element.

An example of a transcendental branch point is the origin for the multi-valued function