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\title{Mathematics 311:  Complex Analysis --- Assignment 8}
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{\bf Reading}:  Marsden, sections 5.1, 5.2.

\bigskip

{\bf Problems}:

1.  Consider a nonidentity fractional linear transformation
$$
Sz=\left[\begin{matrix}a&b\\ c&d\end{matrix}\right](z),
\quad\left[\begin{matrix}a&b\\ c&d\end{matrix}\right]\in{\rm PSL}_2(\C).
$$
Show that $S$ can be classified as elliptic, parabolic, hyperbolic, or
loxodromic solely in terms of the projective trace $\pm(a+d)$, which is
well defined.  For example, $S$ is elliptic if and only if $a+d$ is real
and $|a+d|<2$.

\medskip

2.  The {\em cross-ratio\/} of any four distinct points
$z_1, z_2, z_3, z_4 \in\C$ is defined as
$$
(z_1,z_2,z_3,z_4)=\frac{z_1-z_3}{z_1-z_4}\cdot\frac{z_2-z_4}{z_2-z_3}.
$$

(a) Show that $(z_1,z_2,z_3,z_4)=Sz_1$, where $S$ is the unique fractional
linear transformation such that $Sz_2=1$, $Sz_3=0$, $Sz_4=\infty$.

(b) Use part (a) to show that $(z_1,z_2,z_3,z_4)=(Tz_1,Tz_2,Tz_3,Tz_4)$ 
for any fractional linear transformation $T$.

(c) Use part (b) to show that $(z_1,z_2,z_3,z_4)$ is real precisely
when these four points lie either on the same line or the same circle.

\medskip

3.  (a) Show that the reflection $z^*$ of the point $z$ in the circle passing
through three distinct points $z_1$, $z_2$, $z_3$ is characterized by the
condition
$$
(z^*,z_1,z_2,z_3)=\overline{(z,z_1,z_2,z_3)}.
$$
(Hint for one direction:  If the circle through $z_1$, $z_2$, $z_3$ is centered
at $a$ and has radius $R$ then using properties of the cross-ratio shows that
\begin{align*}
\overline{(z,z_1,z_2,z_3)}
&=\overline{(z-a,z_1-a,z_2-a,z_3-a)}\\
&=(\overline{z}-\overline{a},R^2/(z_1-a),R^2/(z_2-a),R^2/(z_3-a))\\
&=(R^2/(\overline{z}-\overline{a}),z_1-a,z_2-a,z_3-a)\\
&=(R^2/(\overline{z}-\overline{a})+a,z_1,z_2,z_3).
\end{align*}
Since I am giving you the calculation, explain the steps.)

(b) Suppose that $z^*$ is the reflection of $z$ in a circle $C$.  Show that
$Sz^*$ is the reflection of $Sz$ in the circle $SC$.

\medskip

4.  Find fractional linear transformations:

(a) $R$, taking the half-disk $\{z\in\C:|z|<1,\text{Im}(z)>0\}$ to the first
quadrant $\{z\in\C:\text{Re}(z)>0,\text{Im}(z)>0\}$.

(b) $S$, taking the circle $|z|=2$ to the circle $|z+1|=1$, the point $-2$ to
the origin, and the origin to $i$.

(c) $T$, taking the circles $|z|=1$ and $|z-1/4|=1/4$ to $|z|=R$ and
$|z|=1$ respectively, with $R>1$, and taking $1/2\mapsto1$.  What must $R$ be?

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