2.1 The Extended Complex Plane and its Spherical Representation

All complex numbers form a field that extends the real number field. A complex number can be visualized on a rectangular plane as the point , with two axes: the real axis and the imaginary axis. It is well known that any complex number also has the polar form .

The point at infinity, , extends to

The following arithmetic operations are defined: for all ,

and for all ,

Let

There exists a stereographic projection of onto . For every point other than , there is a corresponding complex number

This correspondence between and is injective. In fact, the inverse can be solved for:

which results in

and consequently,

By letting correspond to , the bijection is complete. The sphere is also called the Riemann sphere. The region given by the disk corresponds to , and the region corresponds to .

We will now give a geometric visualization of this projection. Let . Then we obtain

Therefore,

It follows that the points , , and are collinear in . Under the linear map

we get that , , and are collinear. In other words, this correspondence is a central projection with center , projecting the points from onto . Let this center correspond to . In this representation, is no longer considered to be “special”.

It is worth noting that in several geometric contexts, an alternative paradigm exists where we let be the sphere centered at of diameter , and project points from the north pole onto the horizontal plane of tangency. Later in Nevanlinna theory, specifically in @ sec:nevanlinnatheory, we will observe that in some sense this is the more natural object to study. The corresponding equations are then

The forward projection remains unchanged. Lastly, we define the upper half-plane; for the following sections, let