SOFT-JAR-2009 Nº0001 2009 BRANCH POINTS (NOTES) AVALOS´ RODR´IGUEZ JESUS´ A Abstract. This notes are about the some topic in Complex Analysis. It contains about branch points and some examples. Key Words: Branch point, transcendental branch points. 1. Introduction. 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 surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, tran- scendental 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 solu- tion 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. 2. BRANCH POINTS. 2.1. Algebraic branch points. Let Ω be a connected open set in the complex plane C and f : Ω C a holomorphic function. If f is not constant, then the → Date: February 10th, of 2009. A. E-mail address: jeshua [email protected]. 1 2 J. AVALOS´ RODR´IGUEZ critical points of f, that is, the zeros of the derivative f 0(z), have no limit point in Ω. So each critical point z0 of f lies at the center of a disc B(z0, r) containing no other critical point of f in its closure. Let γ be the boundary of B(z0, r), taken with its positive orientation. The winding number of f(y) with respect to the point f(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 f, and the corresponding critical value f(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 f(z)= ϕ(z)(z k − z0) for some positive integer k> 1. Typically, one is not interested in f itself, but in its inverse function. However, the inverse of a holomorphic function in the neighborhood of a branch 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 −1 point z0 of f as a branch point of the global analytic function f . 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 branch points. In terms of the inverse global analytic function f −1, branch points are those points around which there is nontrivial monodromy. 2 For example, the function f(z) = z has a branch point at z0 = 0. The inverse function is the square root f −1(z) = z1/2. Going around the closed loop z = eiθ, one starts at θ = 0 and e(0)i/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. − 2.2. Transcendental and Logarithmic branch Points. Suppose that g is a global analytic function defined on a punctured disc around z0. Then g is said to be 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 g(z) = exp(z)1/k for some integer k > 1. 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. 2.3. Examples. 1. 0 is a branch point of the square root function. Suppose w = √z, and z starts at 4 and moves along a circle of radius 4 in the complex plane 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. 2. 0 is also a branch point of the natural logarithm. 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. BRANCH POINTS (NOTES) 3 3. In trigonometry, 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 πi are branch points of the arctangent function, 1 log( i ) arctan(z)= πz . 2i (i + z) This may be seen by observing that the derivative d 1 arctan(z)= dz 1+ z2 has simple poles at those two points, since the denominator is zero at those points. 4. If the derivative f 0 of a function f has a simple pole at a point a, then f has a branch point at a. (The converse is false, since the square-root function is a counterexample.) 2.4. 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 F (z)= √z√1 z − 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 (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. 2.5. 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 ln(z) = ln(r)+ iθ 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 is undefined on the branch cut, and has a jump discontinuity of 2πi when crossing it. The logarithm can be made continuous by gluing together 4 J. AVALOS´ RODR´IGUEZ Figure 1. A plot of the multi-valued imaginary part of the com- plex logarithm function, which shows the branches. As a complex number z goes around the origin, the imaginary part of the loga- rithm goes up or down. 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.