18.5 The Logarithm and the Problem of the Multivalued Nature of Angles

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

$$\mathrm{ln}z=\mathrm{ln}r+i\theta $$

This formula has a problem, and that problem is that the angle
$\theta $
is not a well defined function in the complex plane, and so neither is
$\mathrm{ln}z$
.

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\pi $
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
$-\pi $
to
$\pi $
. If you do that the function
$\mathrm{ln}z$
is discontinuous on the negative real axis.

Similar problems exists for inverse powers such as
${x}^{1/2}$
and
${x}^{1/3}$
as well.

There are several ways to get around this problem.

The prosaic way is to define such inverse functions precisely by introducing
a line of discontinuity for them, called a cut.

Thus for the logarithm you can say that its imaginary part,
$\theta $
, has values that run from 0 to
$2\pi $
. If so it is discontinuous on the positive real axis, being 0 on one side of it and
$2\pi $
on the other.

Alternatively you can have its values lie from
$-\pi $
to
$\pi $
, so that its line of discontinuity is the negative real axis, and you can choose any other half line of discontinuity starting at the origin.

Another way to handle this problem is to replace the complex plane by a geometric structure called a
Riemann surface,
on which the function in question is single valued without a discontinuity.

In the case of the logarithm this surface winds around and around the origin. For the square root if you go around the origin twice you come back to where you started from.

Exercises:

18.4 Define an appropriate system of cuts for the function${({z}^{2}-1)}^{1/2}$
.
Can you find a definition whose cut is a single line segment?

18.5 Describe the Reimann surface for this function in each case.