Mobius transformation satisfying certain properties I'm having some trouble showing that

Question

Mobius transformation satisfying certain properties
I'm having some trouble showing that a Mobius transformation F maps 0 to

\infty and \infty

to 0 iff

F (z) = {d z}^{- 1}

for some

d \in C

. Mainly with the "only if" part. Do I need to use pictures?

Answer 1

Write

f (z) = \frac{a z + b}{c z + d}

. What are f(0) and

f (\infty)

in terms of a,b,c,d? Deduce further.
(Its

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