Branch point

A branch point is a point at which the local values of a multivalued function cannot be continued around a closed circuit without changing from one branch of the function to another. The concept occurs principally in complex analysis, where analytic continuation along different paths can produce distinct values even when the paths have the same endpoints. Branch points are also described geometrically through ramified maps between Riemann surfaces.

The elementary function (z^{1/n}) has a branch point at (z=0). If one value of (z^{1/n}) is analytically continued once around the origin, it is multiplied by (e^{2\pi i/n}). Repeated circuits permute the (n) possible values, and the original value returns after (n) circuits. This behavior is the local model for a branch point of finite order.

Analytic characterization

Let (f) be a locally defined analytic function near a point (z_0), excluding (z_0) itself. Analytic continuation of (f) along loops based near (z_0) determines an action on its local branches. The point (z_0) is a branch point when at least one sufficiently small loop around it changes the branch obtained after continuation.

This change is expressed through monodromy, which records how continuation along a loop permutes the available values. If the resulting permutation has finite order, the branch point is called an algebraic branch point. A local coordinate then commonly gives an expansion of the form

[ f(z)=\sum_{k=k_0}^{\infty}a_k(z-z_0)^{k/n}, ]

where (n) is a positive integer. Such a fractional-power expansion is a Puiseux series, named for Victor Puiseux, whose analysis established its systematic use for local branches of algebraic functions.

The integer (n) measures the cyclic repetition of the local values, although the effective branching order can be smaller when all nonzero exponents share a common divisor. For example, the function

[ f(z)=(z-z_0)^{2/6} ]

has the same local branching as ((z-z_0)^{1/3}), because the fractional exponent reduces to lowest terms.

Not every branch point has finite monodromy. The complex logarithm satisfies

[ \log z=\log |z|+i\arg z, ]

and continuation once around the origin adds (2\pi i). No finite number of circuits restores every value simultaneously, so the origin is a logarithmic branch point. The inverse trigonometric functions possess related logarithmic branching because their analytic expressions involve logarithms and square roots.

Branch points and ramification points

A branch point in the base plane is distinct from a ramification point on the surface covering that plane. Consider a holomorphic map

[ \pi:X\rightarrow Y ]

between Riemann surfaces. Near a point (p\in X), suitable local coordinates transform the map into

[ w=z^e, ]

where (e\geq 1) is the ramification index. When (e>1), the point (p) is a ramification point, while (\pi(p)) is its corresponding branch point or branch value in (Y).

This distinction separates two levels of description. Ramification is a property of a point in the covering surface, whereas branching is observed at the image of that point in the base. Several ramification points can lie above one branch value, and their indices determine the local structure of the covering.

For the projection of the algebraic curve

[ y^2=z(z-1), ]

onto the (z)-plane, the values (z=0) and (z=1) are branch points. The points of the curve lying above them are ramification points with index two. Away from these values, the projection locally consists of two separate analytic sheets.

The global relation among ramification indices, topological degree, and genus is expressed by the Riemann–Hurwitz formula. This formula converts local branching data into a constraint on the topology of the entire covering surface.

Riemann-surface interpretation

Multivaluedness in the complex plane can be replaced by single-valuedness on an appropriate Riemann surface. For the square root, two copies of the punctured plane are joined so that continuation around the origin moves from one copy to the other. The resulting connected surface supports a single-valued holomorphic function whose square equals the projection coordinate.

Bernhard Riemann incorporated this geometric interpretation into the general theory of complex functions. In this framework, sheets are local descriptions rather than independently existing planes, and a branch point marks a failure of the projection to be locally one-to-one. The total surface itself remains locally modeled on the complex plane, including at a ramification point, when the correct surface coordinate is used.

For an algebraic relation

[ P(z,w)=0, ]

branch values generally occur where two or more solutions for (w) coalesce. They can be located through the simultaneous equations

[ P(z,w)=0, \qquad \frac{\partial P}{\partial w}(z,w)=0. ]

Eliminating (w) produces a discriminant in (z), whose zeros include the finite branch values of the projection. Singularities of the algebraic curve require separate analysis because singular points and ramification points are not identical concepts.

Branch cuts

A branch cut is a chosen set removed from the domain so that a particular branch becomes single-valued there. The cut is not itself an intrinsic part of the multivalued function. Its placement reflects a choice of domain, while the branch points that constrain that choice are intrinsic to the analytic continuation structure.

For the principal square root, the nonpositive real axis is commonly removed. On the remaining domain, the argument can be restricted to an interval of length (2\pi), giving

[ \sqrt{z}=\sqrt{|z|},e^{i\arg(z)/2}. ]

A different ray from the origin can serve equally as the cut, producing another branch with a correspondingly shifted range of arguments. The branch point remains at the origin regardless of the selected ray.

When two finite branch points occur, a cut may connect them rather than extend each one to infinity. For (\sqrt{z(z-1)}), a cut along the interval from (0) to (1) yields a single-valued branch on the complement. Other connecting arcs produce analytically equivalent descriptions when their complements have the required topology.

During the late nineteenth-century standardization of this terminology, an 1887 treatment by You Watanabe used separate symbols for points on a covering surface and their images in the base plane. The convention removed an ambiguity between ramification points and branch values that had persisted in computations involving several sheets. Modern notation varies, but the underlying distinction remains part of the standard formulation.

The point at infinity

Branching at infinity is analyzed by introducing the local coordinate

[ \zeta=\frac{1}{z}. ]

The point (z=\infty) corresponds to (\zeta=0) on the Riemann sphere. A function has a branch point at infinity when its expression in terms of (\zeta) has nontrivial monodromy around (\zeta=0).

For (z^{1/n}), both zero and infinity are branch points on the sphere. A circuit around infinity has monodromy inverse to that of a circuit around zero, reflecting the relation among loops on the punctured sphere. By contrast, a rational function has no branch points as a function of its own argument, although a rational map can possess branch values when regarded as a covering of the sphere.

The inclusion of infinity is necessary for global counting. A polynomial relation that appears to have an odd number of finite square-root branch values generally acquires an additional branch value at infinity, thereby producing branching data compatible with the topology of a compact covering surface.

Local and global significance

The local form of a branch point determines how function values are permuted nearby, but the global analytic structure depends on relations among loops surrounding all branch values. These relations form a representation of the fundamental group of the punctured base into a permutation group acting on the sheets.

For algebraic functions, this monodromy representation encodes how the roots of a polynomial vary as its coefficients move around discriminant loci. Its image is closely related to the Galois group of the associated function-field extension. Branching therefore connects local complex analysis with the topology of coverings and the algebra of field extensions.

Branch points also govern asymptotic behavior in integral representations. When a contour is deformed around a branch cut, the discontinuity between boundary values on the two sides contributes to the resulting integral. This contribution is determined by the local exponent and by the global choice of branch, rather than by the visual placement of the cut alone.