# Riemann Sphere

## The Point at infinity

For some purposes it is convenient to introduce the *point at infinity*, denoted by $\infty,$
in
addition to the points $z\in \mathbb C.$ We must be careful in doing so, because it can lead to
confusion
and abuse of the symbol $\infty.$ However, with care it can be useful, if we want
to be able to talk about infinite limits and limits at infinity.

In contrast to the real line, to which $+\infty$ and $-\infty$ can be added, we have only one
$\infty$ for
$\mathbb C.$ The reason is that $\mathbb C$ has no natural ordering as $\mathbb R$ does. Formally we
add a
symbol $\infty$ to $\mathbb C$ to obtain the *extended complex plane*, denoted by $\mathbb
C^*=\mathbb C \cup \{\infty\},$ and define operations with $\infty$ by the rules

The extended complex plane can be mapped onto the
surface of a sphere whose south pole corresponds to the origin and whose north
pole to the point $\infty.$ All other points of the complex plane can be mapped in
a one-to-one fashion to points on the surface of the sphere by using the following
construction. Connect the point $z$ in the plane with the north pole using a
straight line. This line intersects the sphere at the point $P(z).$ In this way each
point $z= x+iy$ on the complex plane corresponds uniquely to a point $P(z)$ on the
surface of the sphere. This construction is called the *stereographic projection* and is
illustrated
in the following applet.

In the following applet we can observe the unit sphere whose south pole corresponds to the origin of the $z$ plane. Drag the point defined on the $z$ plane, or the sliders, to explore the behaviour of the point $P(z)$ on the sphere.

The extended complex plane is sometimes referred to as the *compactified* (closed) complex
plane. It
is often useful to view the complex plane in this way, and knowledge of the construction of the
stereographic projection is valuable in certain advanced treatments.

## Infinite Limits

Now we can introduce the following limit concepts:

- $\displaystyle \lim_{z\rightarrow \infty}f(z)=z_0$ means: For any $\varepsilon>0,$ there is an $R>0$ such that $$|z|> R \quad \Rightarrow \quad \left|\,f(z)-z_0\right| < \varepsilon.$$
- $\displaystyle \lim_{z\rightarrow z_0}f(z)=\infty$ means: For any $R>0,$ there is a $\delta>0$ such that $$0 \lt |z-z_0|< \delta \quad \Rightarrow \quad |\,f(z)|>R.$$
- $\displaystyle \lim_{z\rightarrow \infty}f(z)=\infty$ means: For any $M>0,$ there is an $R>0$ such that $$|z|> R \quad \Rightarrow \quad |\,f(z)|>M .$$

**Example 1:** If $f(z)=\dfrac{1}{z^2},$ for $z\neq 0,$ then
$$\lim_{z\rightarrow \infty}f(z)=0.$$
In fact, given $\varepsilon>0$ we have
$$\left|\frac{1}{z^2}-0\right|=\frac{1}{\left|z^2\right|}=\frac{1}{\left|z\right|^2}
<\varepsilon$$
by taking $$\left|z\right|>\frac{1}{\sqrt{\varepsilon}}=R.$$

**Example 2:**
Let $f(z)=\dfrac{1}{z-3},$ for $z\neq 3.$ Then
$$\lim_{z\rightarrow 3}f(z)=\infty.$$
In fact, for any given $R>0$ the inequality
$$\frac{1}{\left|z-3\right|}>R$$
holds whenever
$$0<\left|z-3\right| < \frac{1}{R}=\delta.$$

**Example 3:**
Now let $f(z) = \dfrac{2z^3-1}{z^2+1},$ for $z\neq -1.$ Then
\[
\lim_{z\to \infty} f(x) = \infty.
\]
There are different ways to prove this limit.
A first attempt is to find the right inequality.
First, let's begin assuming that $|z|\gt 1.$
This implies the following inequalities:

Now we can write the proof in short.

*Proof: * For any $M\gt0,$ choose $R = 1+ 2M$ such that $|z|\gt R.$ Then

**Remark:** By the way the choice $R=1+2M$
is obviously not unique. We just need $R\geq 1$
to ensure the second inequality in the proof above, and
$R\geq 2M$
to ensure the final inequality.
Even if you end up
with $|f(z)|>M/2$
at the end, that is an equivalent
definition since we have a universal quantifier “for any $M\gt 0$”.

There is an easier way to calculate the limits from Examples 1-3. The following theorem provides a very useful method.

Using this result, we can easily find that

Although we have successfully used techniques from calculus to compute complex-valued limits, real-variable intuition may not always apply. For example, $\lim_{z\to \infty} e^{-z}$ is not 0; in fact, this limit does not exist.