» Required Reading » Table of Contents » Chapter 18 18.5 The Logarithm and the Problem of the Multivalued Nature of Angles
	Previous Section	Next Chapter

The logarithm of z is, from the last equation, describable as

This formula has a problem, and that problem is that the angle is not a well defined function in the complex plane, and so neither is ln z. The difficulty is that as we wander around the origin in a counterclockwise direction, the angle keeps increasing and comes back after each revolution 2 greater than it was. Thus the value of the logarithm at a given value of z depends on how you got there; unless you artificially restrict its angle say to range from - to . If you do that the function ln z is discontinuous on the negative real axis.

This problem exists for inverse powers such as x^1/2 as well.