Notes - Complex Analysis MT23, Extended complex plane
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 $S : \mathbb C _ \infty \to \mathbb S$?
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 $ \vert \vert \cdot \vert \vert $ 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) = \vert \vert S(z) - S(w) \vert \vert $. Can you give an explicit formula for this metric?
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) = \vert \vert S(z) - S(w) \vert \vert $. What is $d(z, \infty)$?
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 $ \vert z - w \vert \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?
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) = \vert \vert S(z) - S(w) \vert \vert $. 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 $a^2 + b^2 + c^2 = 1$.