Branch point
Encyclopedia
In the mathematical
field of complex analysis
, a branch point of a multi-valued function
(usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point . 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 z = w2 for w as a function of 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 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.
in the complex plane
C and ƒ:Ω → C a holomorphic function
. If ƒ is not constant, then the set of the critical point
s of ƒ, that is, the zeros of the derivative ƒ' (z), has no limit point
in Ω. So each critical point z0 of ƒ lies at the center of a disc B(z0,r) containing no other critical point of ƒ in its closure.
Let γ be the boundary of B(z0,r), taken with its positive orientation. The winding number
of ƒ(γ) with respect to the point ƒ(z0) is a positive integer called the ramification index of z0. If the ramification index is greater than 1, then z0 is called a ramification point of ƒ, and the corresponding critical value
ƒ(z0) is called an (algebraic) branch point. Equivalently, z0 is a ramification point if there exists a holomorphic function φ defined in a neighborhood of z0 such that ƒ(z) = φ(z)(z − z0)k for some positive integer k > 1.
Typically, one is not interested in ƒ 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 w0 = ƒ(z0) of ƒ as a branch point of the global analytic function ƒ−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 surface
s 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 ƒ−1, branch points are those points around which there is nontrivial monodromy
. For example, the function ƒ(z) = z2 has a ramification point at z0 = 0. The inverse function is the square root ƒ−1(w) = w1/2, which has a branch point at w0 = 0. Indeed, going around the closed loop w = eiθ, one starts at θ = 0 and ei0/2 = 1. But after going around the loop to θ = 2π, one has e2πi/2 = −1. Thus there is monodromy around this loop enclosing the origin.
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
for some integer k > 1. Here the monodromy around the origin is finite.
By contrast, the point z0 is called a logarithmic branch point if it is impossible to return to the original function element by analytic continuation along a curve with nonzero winding number about z0. 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. Encircling a loop with winding number w, the logarithm is incremented by 2πi w.
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. Such covers are therefore always unramified.
Branch cuts allow one to work with a collection of single-valued functions, "glued" together along the branch cut instead of a multivalued function. 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 same idea can be applied to the function √z; but in that case one has to perceive that the point at infinity is the appropriate 'other' branch point to connect to from 0, for example along the whole negative real axis.
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 function
s and differential equation
s.
The typical example of a branch cut is the complex logarithm. 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. A branch of the logarithm is a continuous function L(z) giving a logarithm of z for all z in a connected open set in the complex plane. In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a branch cut. A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience.
The logarithm has a jump discontinuity of 2πi when crossing the branch cut. The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut. On each sheet, the value of the log differs from its principal value by a multiple of 2πi. 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, a branch point of a multi-valued function
Multivalued function
In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
(usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point . Multi-valued functions are rigorously studied using Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
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 z = w2 for w as a function of z. Here the branch point is the origin, because the analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
. Despite the algebraic branch point, the function 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
Essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...
. In geometric function theory
Geometric function theory
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.-Riemann mapping theorem:...
, 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.
Algebraic branch points
Let Ω be a connected open setOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
C and ƒ:Ω → C a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
. If ƒ is not constant, then the set of the critical point
Critical point (mathematics)
In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
s of ƒ, that is, the zeros of the derivative ƒ
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
in Ω. So each critical point z0 of ƒ lies at the center of a disc B(z0,r) containing no other critical point of ƒ in its closure.
Let γ be the boundary of B(z0,r), taken with its positive orientation. The winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
of ƒ(γ) with respect to the point ƒ(z0) is a positive integer called the ramification index of z0. If the ramification index is greater than 1, then z0 is called a ramification point of ƒ, and the corresponding critical value
Critical value
-Differential topology:In differential topology, a critical value of a differentiable function between differentiable manifolds is the image ƒ in N of a critical point x in M.The basic result on critical values is Sard's lemma...
ƒ(z0) is called an (algebraic) branch point. Equivalently, z0 is a ramification point if there exists a holomorphic function φ defined in a neighborhood of z0 such that ƒ(z) = φ(z)(z − z0)k for some positive integer k > 1.
Typically, one is not interested in ƒ itself, but in its inverse function
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
. 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
Global analytic function
In the mathematical field of complex analysis, a global analytic function is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analytic functions arise naturally in considering the possible analytic continuations of an analytic...
. It is common to abuse language and refer to a branch point w0 = ƒ(z0) of ƒ as a branch point of the global analytic function ƒ−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
Implicit function
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...
. A unifying framework for dealing with such examples is supplied in the language of Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s 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 ƒ−1, branch points are those points around which there is nontrivial monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
. For example, the function ƒ(z) = z2 has a ramification point at z0 = 0. The inverse function is the square root ƒ−1(w) = w1/2, which has a branch point at w0 = 0. Indeed, going around the closed loop w = eiθ, one starts at θ = 0 and ei0/2 = 1. But after going around the loop to θ = 2π, one has e2πi/2 = −1. Thus there is monodromy around this loop enclosing the origin.
Transcendental and logarithmic branch points
Suppose that g is a global analytic function defined on a punctured discAnnulus (mathematics)
In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
around z0. Then g has a transcendental branch point if z0 is an essential singularity
Essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...
of g such that analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
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
for some integer k > 1. Here the monodromy around the origin is finite.
By contrast, the point z0 is called a logarithmic branch point if it is impossible to return to the original function element by analytic continuation along a curve with nonzero winding number about z0. This is so called because the typical example of this phenomenon is the branch point of the complex logarithm
Complex logarithm
In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...
at the origin. Going once counterclockwise around a simple closed curve encircling the origin, the complex logarithm is incremented by 2πi. Encircling a loop with winding number w, the logarithm is incremented by 2πi w.
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. Such covers are therefore always unramified.
Examples
- 0 is a branch point of the square rootSquare rootIn mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
function. Suppose w = z1/2, and z starts at 4 and moves along a circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
of radiusRadiusIn classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
4 in the complex planeComplex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
centered at 0. The dependent variable w changes while depending on z in a continuous manner. When z has made one full circle, going from 4 back to 4 again, w will have made one half-circle, going from the positive square root of 4, i.e., from 2, to the negative square root of 4, i.e., −2.
- 0 is also a branch point of the natural logarithmNatural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
. Since e0 is the same as e2πi, both 0 and 2πi are among the multiple values of Log(1). As z moves along a circle of radius 1 centered at 0, w = Log(z) goes from 0 to 2πi.
- In trigonometryTrigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
, since tan(π/4) and tan (5π/4) are both equal to 1, the two numbers π/4 and 5π/4 are among the multiple values of arctan(1). The imaginary units i and −i are branch points of the arctangent function (arctan(z) = (1/2i)log(i − z)/(i + z)). This may be seen by observing that the derivative (d/dz) arctan(z) = 1/(1 + z2) has simple poles at those two points, since the denominator is zero at those points.
- If the derivative ƒ
' of a function ƒ has a simple pole at a point a, then ƒ has a logarithmic branch point at a. The converse is not true, since the function ƒ(z) = zα for irrational α has a logarithmic branch point, and its derivative is singular without being a pole.
Branch cuts
Roughly speaking, branch points are the points where the various sheets of a multiple valued function come together. The branches of the function are the various sheets of the function. For example, the function w = z1/2 has two branches: one where the square root comes in with a plus sign, and the other with a minus sign. A branch cut is a curve in the complex plane such that it is possible to define a single branch of a multi-valued function. Branch cuts are usually, but not always, taken between pairs of branch points.Branch cuts allow one to work with a collection of single-valued functions, "glued" together along the branch cut instead of a multivalued function. 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 same idea can be applied to the function √z; but in that case one has to perceive that the point at infinity is the appropriate 'other' branch point to connect to from 0, for example along the whole negative real axis.
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
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
theory (of which it is historically the origin), and more generally in the ramification
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
and monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
theory of algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
s and differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s.
Complex logarithm
The typical example of a branch cut is the complex logarithm. 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. A branch of the logarithm is a continuous function L(z) giving a logarithm of z for all z in a connected open set in the complex plane. In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a branch cut. A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience.
The logarithm has a jump discontinuity of 2πi when crossing the branch cut. The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut. On each sheet, the value of the log differs from its principal value by a multiple of 2πi. 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.
Continuum of poles
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. For example,-
is a function with a simple pole at z = a. Integrating over the location of the pole:
-
defines a function u(z) with a cut from −1 to 1. The branch cut can be moved around, since the integration line can be shifted without altering the value of the integral so long as the line does not pass across the point z.
Riemann surfaces
The concept of a branch point is defined for a holomorphic function ƒ:X → Y from a compact connected Riemann surfaceRiemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
X to a compact Riemann surface Y (usually the Riemann sphereRiemann sphereIn mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
). Unless it is constant, the function ƒ will be a covering map onto its image at all but a finite number of points. The points of X where ƒ fails to be a cover are the ramification points of ƒ, and the image of a ramification point under ƒ is called a branch point.
For any point P ∈ X and Q = ƒ(P) ∈ Y, there are holomorphic local coordinatesLocal coordinatesLocal coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...
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. But if the ramification index is not equal to one, then P is by definition a ramification point, and Q is a branch point.
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 γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is
This integral is the number of times ƒ(γ) winds around the point Q. As above, P is a ramification point and Q is a branch point if eP > 1.
Algebraic geometry
In the context of algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the notion of branch points can be generalized to mappings between arbitrary algebraic curveAlgebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s. Let ƒ:X → Y be a morphism of algebraic curves. By pulling back rational functions on Y to rational functions on X, K(X) is a field extensionField extensionIn abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of K(Y). The degree of ƒ is defined to be the degree of this field extension [K(X):K(Y)], and ƒ is said to be finite if the degree is finite.
Assume that ƒ is finite. For a point P ∈ X, the ramification index eP is defined as follows. Let Q = ƒ(P) and let t be a local uniformizing parameterLocal parameterIn the geometry of complex algebraic curves, a local parameter for a curve C at a smooth point P is just a meromorphic function on C that has a simple zero at P...
at P; that is, t is a regular function defined in a neighborhood of Q with t(Q) = 0 whose differential is nonzero. Pulling back t by ƒ defines a regular function on X. Then
where vP is the valuationValuation ringIn abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....
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.
-