Question #169784

 Find the mobius transformation which maps the 

circle |𝑧| < 1 conformally onto |𝑤 − 1| < 1 and 

make the points z=0,1 correspond to w=1/2,0 

respectively. Is the transformation uniquely 

determined


1
Expert's answer
2021-03-11T04:21:21-0500

The Mobius transformation is generally given by

f(z)=az+bcz+df(z) = \frac{az+b}{cz+d}

As f(0)f(0)\neq \infty, we conclude that d0d \neq 0 and thus we can choose d=2d=2 (as in Mobius transformation multiplying a,b,c,da,b,c,d by a same non-zero constant gives the same transformation). Knowing the images if z=0,z=1z=0, z=1 we find:

  • bd=12,b=1\frac{b}{d}=\frac{1}{2}, b=1
  • a+b=0,a=1a+b=0, a=-1

Now suppose zC,z=1z\in \mathbb{C}, |z|=1, then this tranformation should carry it onto w1=1|w-1|=1, thus

f(z)1=z+1(cz+2)cz+2=(c+1)z1cz+2|f(z)-1|=|\frac{-z+1-(cz+2)}{cz+2}|=|\frac{-(c+1)z-1}{cz+2}|

f(z)1=(1+(c+1)cosθ)2+(c+1)2sin2θ(ccosθ+2)2+c2sin2θ=1|f(z)-1|=\frac{(1+(c+1)\cos \theta)^2+(c+1)^2\sin^2\theta }{(c\cos\theta +2)^2+c^2 \sin^2\theta}=1

1+(c+1)2cos2θ+2(c+1)cosθ+(c+1)2sin2θ=c2cos2θ+4ccosθ+4+c2sin2θ1+(c+1)^2\cos^2\theta+2(c+1)\cos\theta +(c+1)^2\sin^2\theta =c^2\cos^2\theta+4c\cos\theta +4+c^2\sin^2\theta

1+(1+c)2+2(c+1)cosθ=c2+4ccosθ+41+(1+c)^2+2(c+1)\cos \theta = c^2 +4c\cos\theta+4

(1+2c)+2(1c)cosθ=3(1+2c)+2(1-c)\cos \theta =3

2(1c)cosθ=2(1c)2(1-c)\cos \theta =2(1-c)

As this should be satisfied for all θ[0;2π]\theta \in [0;2\pi], we have c=1c=1 and thus the Mobius transformation is

f(z)=z+1z+2f(z)=\frac{-z+1}{z+2} and the couple [1:1:1:2][-1:1:1:2] is unique up to a multiplication by a scalar.


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