Discussion
The number of Mobius transformations which map the real line onto the unit circle is :
Answer: D
There are infinitely many such Mobius transformations.
One example is the Cayley-type map \(T(z) = \dfrac{z - i}{z + i}\), which sends every real \(z\) to a point with \(|T(z)| = 1\) (real \(z\) is equidistant from \(i\) and \(-i\)) and the real line onto the unit circle.
Composing T with any Mobius map of the real line to itself (for example \(z \mapsto az + b\) with real \(a \ne 0\) and \(b\)), or with any rotation \(w \mapsto e^{i\theta}w\) of the circle, gives another such map. These form a three-parameter real family, so there are infinitely many, not 0, 1 or 2.
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