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The question is, Find an LFT that maps |z|<=1 onto |w|<=1 so that z=i/2 is mapped onto w=0. Sketch the images of the lines x=const and y=const.

By searching and solving, I found several different solutions.

  1. $w=\frac{2}{3}(z-\frac{1}{2})$
  2. $w=\frac{i-2z}{iz+2}$
  3. $w=\frac{z-i/2}{1-iz/2}$

Which one is the right thing?

J sw
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  • Option 1 is not a solution because it operates a shrinking by factor $2/3$ : therefore, the unit disk cannot be mapped onto itself... – Jean Marie Nov 28 '23 at 09:10
  • The two others are good. For the last one, multiply its numerator and denominator by 2 in order to make it look like the other one... – Jean Marie Nov 28 '23 at 09:28
  • @Jean Marie I think the answer is second one. But the last one is slightly different with second one. That makes me little confused. – J sw Nov 28 '23 at 10:11
  • Indeed your sol. (3) doesn't map the unit disk on itself. For example, the image of $\frac{1-i}{2}$ is $1-i$ ! – Jean Marie Nov 28 '23 at 10:32

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