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 Exercise [08.07] 
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Joined: 30 Jun 2008, 22:14
Posts: 25
Post 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 t = \frac{Az + B}{Cz + D} can be re-written as t = \frac{z + B/A}{C/Az + D/A} = \frac{z + b}{cz + d}. So, one needs three linear equations for the unknown b, c and d.
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:
z = -1 \mapsto t = \infty (rotation of north pole to -1)
z = +i \mapsto t = +1 (direction of the t-plane real axis)
It also follows from the first choice that
z = +1 \mapsto t = 0
Substitute these three z and t to the Mobius transformatioin to get three linear equations whose solutions are b = -1 and c = d = i.
Finally, t = \frac{z - 1}{iz + i}. The reverse correspondence, z dependent on t is straightforward.

10 Feb 2009, 23:26
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