Branch Point - Examples

Examples

• 0 is a branch point of the square root function. Suppose w = z1/2, 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 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.