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
$$
L(z) = \frac{z(\alpha w_3-\beta w_1) +
(\beta w_1z_3 - \alpha z_1w_3)}
{z(\alpha-\beta)+(\beta z_3-\alpha z_1)}.
$$
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 $0$.
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 $0$.
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
$$
T(z)=\frac{\lambda}{z}\qquad\text{with }|\lambda|=1.$$
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