Branch point

[1] Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.

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

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

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.

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.

[3] An example of a transcendental branch point is the origin for the multi-valued function for some integer k > 1.

Analytic continuation around k full circuits brings the function back to the original.

If the monodromy group is infinite, that is, it is impossible to return to the original function element by analytic continuation along a curve with nonzero winding number about z0, then the point z0 is called a logarithmic branch point.

[4] This is so called because the typical example of this phenomenon is the branch point of the complex logarithm at the origin.

Going once counterclockwise around a simple closed curve encircling the origin, the complex logarithm is incremented by 2πi.

There is no corresponding notion of ramification for transcendental and logarithmic branch points since the associated covering Riemann surface cannot be analytically continued to a cover of the branch point itself.

A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve.

For example, to make the function single-valued, one makes a branch cut along the interval [0, 1] on the real axis, connecting the two branch points of the function.

The branch cut device may appear arbitrary (and it is); but it is very useful, for example in the theory of special functions.

An invariant explanation of the branch phenomenon is developed in Riemann surface theory (of which it is historically the origin), and more generally in the ramification and monodromy theory of algebraic functions and differential equations.

If a complex number is represented in polar form z = reiθ, then the logarithm of z is However, there is an obvious ambiguity in defining the angle θ: adding to θ any integer multiple of 2π will yield another possible angle.

The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut.

These surfaces are glued to each other along the branch cut in the unique way to make the logarithm continuous.

Each time the variable goes around the origin, the logarithm moves to a different branch.

One reason that branch cuts are common features of complex analysis is that a branch cut can be thought of as a sum of infinitely many poles arranged along a line in the complex plane with infinitesimal residues.

Unless it is constant, the function ƒ will be a covering map onto its image at all but a finite number of points.

For any point P ∈ X and Q = ƒ(P) ∈ Y, there are holomorphic local coordinates z for X near P and w for Y near Q in terms of which the function ƒ(z) is given by for some integer k. This integer is called the ramification index of P. Usually the ramification index is one.

If Y is just the Riemann sphere, and Q is in the finite part of Y, then there is no need to select special coordinates.

The ramification index can be calculated explicitly from Cauchy's integral formula.

Let Q = ƒ(P) and let t be a local uniformizing parameter at P; that is, t is a regular function defined in a neighborhood of Q with t(Q) = 0 whose differential is nonzero.

Then where vP is the valuation in the local ring of regular functions at P. That is, eP is the order to which

vanishes at P. If eP > 1, then ƒ is said to be ramified at P. In that case, Q is called a branch point.