$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\Res}{\operatorname{Res}}$

### Tutorial 11 (Week 12)

Not all examples should be covered.

### Problems

1. Let ${_1,z_2,z_3}$ and ${w_1,w_2,w_3}$ be two triples of distinct complex numbers. Set $$\alpha=\frac{z_2-z_3}{z_2-z_1}\qquad\text{and}\qquad \beta=\frac{w_2-w_3}{w_2-w_1}.$$ Show that the linear fractional transformation that maps $z_j$ to $w_j$ for $j = 1, 2, 3$ is given explicitly by $$L(z) = \frac{z(\alpha w_3-\beta w_1) + (\beta w_1z_3 - \alpha z_1w_3)} {z(\alpha-\beta)+(\beta z_3-\alpha z_1)}.$$
2. Verify that the linear fractional transformation $$S(z) = \frac{z-z_1}{z_2-z_1}$$ maps $z_1$ to $\infty$, $z_2$ to $1$ and $\infty$ to $0$.
3. Verify that the linear fractional transformation $$T(z) = \frac{z_2-z_1}{z-z_1}$$ maps $z_1$ to $\infty$, $z_2$ to $1$, and $\infty$ to $0$.
4. Show that a linear fractional transformation $T$ that maps the circle ${|z| = 1}$ onto itself has the form $$T(z) = \lambda \frac{z-\gamma}{1-\bar{\gamma}z}\qquad\text{with }|\lambda|=1, \ |\gamma|\ne 1$$ or $$T(z)=\frac{\lambda}{z}\qquad\text{with }|\lambda|=1.$$ ----