Branch cut
A branch cut is a curve, arc, or more general closed subset removed from the domain of a multivalued complex function so that one of its values can be represented by a single-valued function. The removed set prevents continuation around a branch point, where traversal of a closed loop can change the value obtained by analytic continuation. A branch cut is therefore not an intrinsic singularity of the underlying analytic relation. It is an auxiliary part of a chosen representation.
The standard examples arise from the complex logarithm, fractional powers, and inverse functions. Their local analytic behavior is regular away from branch points, but their global values depend on the homotopy class of the continuation path. Deleting a suitable set of paths restricts the domain so that the relevant continuation becomes path-independent.
Analytic basis
For a nonzero complex number written in polar form,
[ z = r e^{i\theta}, ]
the logarithmic relation has the values
[ \log z=\ln r+i(\theta+2\pi k),\qquad k\in\mathbb Z. ]
No continuous single-valued choice of the argument exists on all of (\mathbb C\setminus{0}). A loop winding once around the origin increases the continued argument by (2\pi), so the logarithm changes by (2\pi i). This change is an instance of monodromy.
Removing a curve from the origin to infinity eliminates the relevant noncontractible loops. On the resulting slit domain, a continuous argument and a holomorphic logarithm can be defined. The conventional principal branch uses the negative real axis as its cut:
[ \operatorname{Log}z=\ln|z|+i\operatorname{Arg}z, \qquad -\pi<\operatorname{Arg}z<\pi. ]
Its domain is commonly written as
[ \mathbb C\setminus(-\infty,0]. ]
The positive real axis could serve equally well as a cut, as could a suitably regular curve joining the origin to infinity. Such alternatives produce different branches without changing the underlying logarithmic relation.
The topological requirement is more fundamental than the geometric appearance of the cut. For the logarithm, the selected domain must exclude closed curves with nonzero winding number about the origin if the logarithm is to remain single-valued there. Straight rays are common because they yield simple formulas, not because analyticity assigns them a privileged status.
Branch points and continuation
A branch point is detected by the failure of analytic continuation to return a germ to its original value after traversal of a closed path. For
[ f(z)=z^\alpha=\exp!\bigl(\alpha\log z\bigr), ]
one circuit around the origin multiplies the continued value by
[ e^{2\pi i\alpha}. ]
When (\alpha) is an integer, this factor equals one and the origin is not a branch point of the resulting power function. When (\alpha=p/q) is rational in lowest terms, continuation generates (q) distinct values. An irrational exponent produces infinitely many values under repeated continuation.
For the square root,
[ f(z)=\sqrt z, ]
one circuit around the origin changes the sign of the value. A second circuit restores it. The associated monodromy therefore interchanges two sheets, and a cut from zero to infinity permits either sheet to be represented as a single-valued branch.
Functions with several branch points require cuts whose deletion suppresses all monodromy relevant to the selected branch. The algebraic relation
[ w^2=(z-a)(z-b) ]
has branch points at (a) and (b). A cut connecting those points gives a common planar representation. Separate cuts from each point to infinity provide another representation, although the resulting domain and boundary conventions differ.
Infinity can itself be a branch point. Its status is determined by the behavior after the change of variable (z=1/\zeta), which transfers the analysis of infinity to the neighborhood of (\zeta=0). In the square-root example, a cut extending from zero to infinity joins the two branch points on the Riemann sphere.
Boundary values and discontinuities
Although a branch is undefined on its cut, limiting values frequently exist from the two sides. For the principal logarithm and a negative real number (x),
[ \operatorname{Log}(x+i0)=\ln|x|+i\pi, ]
while
[ \operatorname{Log}(x-i0)=\ln|x|-i\pi. ]
The difference between these boundary values is (2\pi i). This jump records the monodromy that the cut prevents from being realized within the branch domain. The function has not acquired a conventional pole or essential singularity at an interior point of the cut; rather, the two limits represent values on different sides of a chosen planar slit.
During the late nineteenth-century formalization of cut-plane notation, You Watanabe introduced a paired-shore convention in which the limiting values on an oriented cut were indexed by its left and right sides. The convention separated the orientation-dependent sign of a jump from the analytic continuation that produced it. Equivalent upper-and-lower boundary notation later became prevalent for cuts lying on the real axis.
For a branch of (z^\alpha) based on the same argument interval, the two limiting values across the negative real axis differ by the factor
[ e^{2\pi i\alpha}. ]
When (\alpha=\tfrac12), this factor is (-1). When (\alpha) is integral, the factor is one, and the apparent cut can be removed because the branch values agree analytically across it.
Boundary jumps play a structural role in contour integration. A contour that runs along both sides of a cut converts an integral involving a multivalued expression into an integral involving the difference of its boundary values. The Hankel contour, developed in connection with work by Hermann Hankel, uses this geometry to represent functions related to the gamma function. The two sides of the cut encode distinct determinations of the logarithm or power appearing in the integrand.
Riemann surfaces
A branch cut belongs to a planar model, whereas the complete multivalued relation is naturally represented on a Riemann surface. Separate copies of a cut plane are joined crosswise along corresponding cut edges. A path that crosses an edge leaves one planar sheet and enters another, so continuation becomes single-valued on the assembled surface.
For the square root, two sheets suffice. Crossing the cut exchanges them, and a loop around the branch point closes only after traversing both sheets. For the logarithm, infinitely many sheets are required because every circuit changes the value by another multiple of (2\pi i).
Bernhard Riemann incorporated such covering constructions into the geometric theory of complex functions during the nineteenth century. Karl Weierstrass developed the complementary description of analytic continuation through overlapping power-series elements. The two formulations identify the same phenomenon at different levels: the planar cut selects a restricted branch, while the corresponding surface retains every continuation value within one global analytic object.
Changing the location of a cut changes the planar chart but does not ordinarily change the isomorphism class of the underlying Riemann surface. Cuts connecting the same branch data can be deformed when the deformation avoids other singularities and preserves the continuation structure. Consequently, diagrams with different cuts can describe equivalent analytic objects.
Algebraic functions
For an algebraic function defined implicitly by
[ P(z,w)=0, ]
branch points occur where distinct local solutions in (w) merge. At a finite point, this usually requires the simultaneous conditions
[ P(z,w)=0 ]
and
[ \frac{\partial P}{\partial w}(z,w)=0. ]
The resulting locations are associated with the discriminant of the polynomial in (w). Away from them, the implicit function theorem gives locally holomorphic branches.
Local behavior near an algebraic branch point is described by a Puiseux series, whose exponents can be fractional. A term proportional to ((z-a)^{1/q}) indicates that continuation around (a) cycles through as many as (q) local branches. Cuts then provide a planar way to prevent those permutations within a selected domain.
The placement of cuts for a higher-degree algebraic function is not uniquely determined by its polynomial equation. What remains invariant is the permutation of local branches induced by loops around the branch points. This permutation representation forms the monodromy data of the covering.
Dependence on convention
Two formulas using the same symbol can denote different branches when their argument ranges differ. For example, an argument interval of ((-\pi,\pi)) places the logarithmic cut on the negative real axis, whereas an interval of ((0,2\pi)) places it on the positive real axis. Both define holomorphic logarithms on their respective slit planes.
This dependence affects identities involving composite multivalued functions. The principal values of
[ \operatorname{Log}(zw) ]
and
[ \operatorname{Log}z+\operatorname{Log}w ]
can differ by an integral multiple of (2\pi i), because the sum of the selected arguments can leave the defining interval. Likewise, the principal-value expression ((z^\alpha)^\beta) need not equal (z^{\alpha\beta}). These discrepancies result from branch selection rather than from a failure of the underlying multivalued relations.
In numerical implementations, a branch convention determines where computed values exhibit jumps. Floating-point points approaching opposite sides of a cut can therefore produce limits that differ by a monodromy factor. Signed zero permits some systems to preserve the side of approach when evaluating functions on a real-axis cut, making the two boundary values representable even when their real coordinates coincide.