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Exercise [08.07]
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Joined: 30 Jun 2008, 22:14
Posts: 25
Exercise [08.07]
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.

10 Feb 2009, 23:26
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