See text and/or instructor's solution manual.
the point onto
the point .
For the function we have . Since ,
by Theorem 10.1 in Section 10.1 , the mapping is conformal at and "angles are preserved."
ray at make
an angle . In
10.2, we proved that a bilinear transformation maps the
class of half-planes and disks onto itself.
Hence, the image of a line through the origin under the Möbius transformation is a "circle." Since , and
the image of the ray will be an arc of a circle that passes through the points .
Therefore, the arc is inclined at the angle at the point .
We are done.
Aside. The image of
will be the point on
the arc as
shown by the calculation
We are really done.
Aside. We can let Mathematica double check our work.
We are really really done.
Aside. We can explore some graphs.
image of the ray under is
an arc of
a circle that passes through and .
If the ray make an angle at then the arc is inclined at the angle at the point .
This solution is complements of the authors.
(c) 2008 John H. Mathews, Russell W. Howell