Branch point

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In the mathematical field of complex analysis, a branch point may be informally thought of as a point z0 at which a "multi-valued function" (usually referred to as a "multifunction" in the context of complex analysis) changes values when one winds once around z0.

Examples:

  • 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.
  • 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.
  • 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 −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 f ' 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.)

In order to work with single-valued functions, it is customary to construct branch cuts in the complex plane, namely arcs out of branch points in the complement of which there is a well-defined branch of the function in question. An example for

F(z) = \sqrt{z} \sqrt{1-z}\,

is to make 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. See also principal branch.

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.