The wording is not entirely clear in this exercise. Somebody who posted a solution in the discussion forum was unsure what is actually asked. My guess is that to find a Mobius transformation that rotates the z-plane unit circle into the t-plane real axis.
A Mobius transformation
can be re-written as
. So, one needs three linear equations for the unknown
,
and
.
Again, there're ambiguities. Any transformation that rotates the north pole to any point on the unit circle will do. Moreover, one can freely choose the direction of the t-plane real axis. Two degrees of freedom, so it seems.
I tested and knew that the following two choices will provide the answer Penrose asked for:
(rotation of north pole to -1)
(direction of the t-plane real axis)
It also follows from the first choice that
Substitute these three
and
to the Mobius transformatioin to get three linear equations whose solutions are
and
.
Finally,
. The reverse correspondence,
dependent on
is straightforward.
