# Notes - Complex Analysis MT23, Extended complex plane

> Source: https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/notes/extended-complex-plane/ · Updated: 2023-09-02 · Tags: uni, notes

- [Course - Complex Analysis MT23](https://ollybritton.com/notes/uni/part-a/mt23/complex-analysis/)

### Flashcards
#### Riemann sphere model
When using the stereographic projection model of $\mathbb C_\infty$, how do we define $\mathbb S$, $N$, and how do we view the complex plane $\mathbb C$?::
$\mathbb S$ is the unit sphere of radius 1, centered at the origin in $\mathbb R^3$, i.e.
$$
\mathbb S = \\{(x, y, z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 1\\}
$$
We view $\mathbb C$ as a copy of the plane $\mathbb R^2$ given by
$$
\\{(x, y, 0) \in \mathbb R : x, y \in \mathbb R\\}
$$
and the $N$ is the “north pole” $N = (0, 0, 1) \in \mathbb S$.

When using the stereographic projection model of $\mathbb C_\infty$, what bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$ do we use, in words?::
Consider a point $z$. Join $z$ to $N$ by a line, and let $S(z)$ be the point where it intersects the sphere $\mathbb S$.

When using the stereographic projection model of $\mathbb C_\infty$, we  use a bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$ that maps complex numbers to points on the sphere. How do we add in $\infty$, to make $S : \mathbb C_\infty \to \mathbb S$?::
$$
S(\infty) = N
$$

When using the stereographic projection model of $\mathbb C_\infty$, we use the following bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$: Consider a point $z$. Join $z$ to $N$ by a line, and let $S(z)$ be the point where it intersects the sphere $\mathbb S$.

Now suppose $z = x + iy$. Can you give the explicit formula for $S(z)$, first in coordinates $(x', y', z')$ and then in its alternative form just in terms of $z$?::
$$
S(z) = \left(\frac{2x}{x^2 + y^2 + 1}, \frac{2y}{x^2 + y^2 + 1}, \frac{x^2 + y^2 - 1}{x^2 + y^2 + 1}\right)
$$
alternatively:
$$
S(z) = \frac{1}{1 + |z|^2} (2 \Re (z), 2\Im(z), |z|^2 -1)
$$

When using the stereographic projection model of $\mathbb C_\infty$, we use a bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$ that maps any point $z$ onto a point of a sphere $\mathbb S$. How do we use this to induce a metric on $\mathbb C$, say $d(z, w)$?::
Define
$$
d(z, w) = ||S(z) - S(w)||
$$
where $||\cdot||$ is the normal Euclidean metric on $\mathbb R^3$.

When using the stereographic projection model of $\mathbb C_\infty$, we use a bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$ that maps any point $z$ onto a point of a sphere $\mathbb S$. We then induce a metric on $\mathbb C$ via $d(z, w) = ||S(z) - S(w)||$. Can you give an explicit formula for this metric?::
$$
d(z, w) = \frac{2|z-w|}{\sqrt{1 + |z|^2} \sqrt{1+ |w|^2}\,}
$$

When using the stereographic projection model of $\mathbb C_\infty$, we use a bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$ that maps any point $z$ onto a point of a sphere $\mathbb S$. We then induce a metric on $\mathbb C$ via $d(z, w) = ||S(z) - S(w)||$. What is $d(z, \infty)$?::
$$
d(z, \infty) = \frac{2}{\sqrt{1 + |z|^2}\,}
$$

When using the stereographic projection model of $\mathbb C_\infty$, we use a bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$ that maps any point $z$ onto a point of a sphere $\mathbb S$. This induces a metric on $\mathbb C$ given by
$$
d(z, w) = ||S(z) - S(w)||
$$
What is special about this metric in relation to the standard metric on $\mathbb C$?::
They are strongly equivalent on any bounded set, and equivalent on the whole of $\mathbb C$.

How (and why) can you show a function $f : \mathbb C_\infty \to \mathbb C_\infty$ that is continuous when restricted to $\mathbb C$ is in fact continuous on all of $\mathbb C_\infty$?::
Examine the continuity at $\infty$ using metric induced from the Riemann sphere, and the rest is handled by the fact these metrics are equivalent on $\mathbb C$.

If $z, w \in \mathbb C_\infty \setminus \\{\infty\\}$, what can you say about $d(z, w) \to 0$ and $|z - w| \to 0$?::
$$
d(z, w) \to 0 \iff |z-w| \to 0
$$

#### Complex projective line model
Can you define $\mathbb P^1(\mathbb C)$ formally?::
The set of equivalence classes of the relation defined on $\mathbb C^2 \backslash \\{0\\}$ as follows: $(z, w) \sim (z', w')$ iff there exists $\lambda \ne 0$ such that $\lambda (z, w) = (z', w')$.

Can you define $\mathbb P^1(\mathbb C)$ intuitvely?::
The set of all (complex) lines passing through the origin.

What notation is used for the equivalence classes of a point $(z, w)$ in $\mathbb P^1(\mathbb C)$, and why?::
$$
[z : w]
$$
this is supposed to be reminiscient of considering it the line with the ratio $z / w$.

In the projective line model of $\mathbb C_\infty$, how do we form a bijective map $\iota$ between $\mathbb C \cup \\{\infty\\}$ and $\mathbb P^1 (\mathbb C)$? Give the formal correspondence and the intuitive correspondence::
- $\iota(z) = [z : 1]$, $z$ is identified with the line passing through $(z, 1)$.
- $\iota(\infty) = [1 : 0]$, $\infty$ is identified with the line with “infinite slope”.

When using the stereographic projection model of $\mathbb C_\infty$, we use a bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$ that maps any point $z$ onto a point of a sphere $\mathbb S$. We then induce a metric on $\mathbb C$ via $d(z, w) = ||S(z) - S(w)||$. Quickly prove that the explicit formula for this metric is given by
$$
d(z, w) = \frac{2|z-w|}{\sqrt{1 + |z|^2} \sqrt{1+ |w|^2}
}
$$
and that
$$
d(z, \infty) = \frac{2}{\sqrt{1 + |z|^2}\\,}
$$

::

Use the fact that
$$
||S(z) - S(w)||^2 = ||S(z)^2|| - 2\langle S(z), S(w)\rangle + ||S(w)||^2
$$
and since $S(z)$ and $S(w)$ are on the unit sphere, this becomes
$$
||S(z) - S(w)||^2 = 2 - 2\langle S(z), S(w)\rangle
$$
Then
$$
\begin{aligned}
\langle S(z), S(w)\rangle &= \frac{1}{1 + |z|^2}\frac{1}{1+|w|^2} \left( 4\Re(z)\Re(w) + 4\Im(z) \Im(w) + (|z|^2 - 1)(|w|^2 - 1) \right) \\\\
&= \frac{4(\Re(z) \Re(w) + \Im(z) \Im(w)) + |z|^2 |w|^2 - |z|^2 - |w|^2 + 1}{(1+|z|^2)(1+|w|^2)} \\\\
&= \frac{4(\Re(z) \Re(w) + \Im(z) \Im(w)) + (1 + |z|^2)(1 + |w|^2) - 2|z|^2 - 2|w|^2}{(1+|z|^2)(1+|w|^2)} \\\\
&= 1 + \frac{4(\Re(z) \Re(w) + \Im(z) \Im(w)) - 2|z|^2 - 2|w|^2}{(1+|z|^2)(1+|w|^2)} \\\\
\end{aligned}
$$
Then noting that
$$
\begin{aligned}
2|z - w|^2 &= 2|\Re(z) + i \Im(z) -\Re(w) - i\Im(w)|^2 \\\\
&= \cdots (\text{trust me}) \\\\
&= -4(\Re(z) \Re(w) + \Im(z) \Im(w)) + 2 |z|^2 + 2|w|^2
\end{aligned}
$$
Then
$$
\begin{aligned}
\langle S(z), S(w)\rangle &= 1 - \frac{2|z-w|^2}{(1+|z|^2)(1+|w|^2)} \\\\
\end{aligned}
$$
so
$$
||S(z) - S(w)||^2 = \frac{4|z-w|^2}{(1 + |z|^2)(1 + |w|^2)} 
$$
as required. The case for $w = \infty$ is easier as $S(w) = (0, 0, 1)$.

### Proofs
When using the stereographic projection model of $\mathbb C_\infty$, we use the following bijective map $S : \mathbb C \to \mathbb S \backslash \\{N\\}$: Consider a point $z$. Join $z$ to $N$ by a line, and let $S(z)$ be the point where it intersects the sphere $\mathbb S$.

Now suppose $z = x + iy$. Prove that the explicit formula for $S(z)$ is given by
$$
S(z) = \left(\frac{2x}{x^2 + y^2 + 1}, \frac{2y}{x^2 + y^2 + 1}, \frac{x^2 + y^2 - 1}{x^2 + y^2 + 1}\right)
$$
::

Not sure if there is a quick way, consider the parameterisation
$$
(1-t)(x, y, 0) + t(0, 0, 1)
$$
and then solve for $a^2 + b^2 + c^2 = 1$.

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt