The Mobius transformation is generally given by
f(z)=cz+daz+b
As f(0)=∞, we conclude that d=0 and thus we can choose d=2 (as in Mobius transformation multiplying a,b,c,d by a same non-zero constant gives the same transformation). Knowing the images if z=0,z=1 we find:
- db=21,b=1
- a+b=0,a=−1
Now suppose z∈C,∣z∣=1, then this tranformation should carry it onto ∣w−1∣=1, thus
∣f(z)−1∣=∣cz+2−z+1−(cz+2)∣=∣cz+2−(c+1)z−1∣
∣f(z)−1∣=(ccosθ+2)2+c2sin2θ(1+(c+1)cosθ)2+(c+1)2sin2θ=1
1+(c+1)2cos2θ+2(c+1)cosθ+(c+1)2sin2θ=c2cos2θ+4ccosθ+4+c2sin2θ
1+(1+c)2+2(c+1)cosθ=c2+4ccosθ+4
(1+2c)+2(1−c)cosθ=3
2(1−c)cosθ=2(1−c)
As this should be satisfied for all θ∈[0;2π], we have c=1 and thus the Mobius transformation is
f(z)=z+2−z+1 and the couple [−1:1:1:2] is unique up to a multiplication by a scalar.
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