Thread: find mobius transformation that maps 3 points

Hi guys,

Here's the question: find the Mobius transformation mapping 0,1,infinity to 1, 1+i, i respectively.



I know can find the transformation mapping 1, 1+i, i to 0,1,infinity using:

f(z) = [(z - z1) / (z - z3)] * [(z2 - z3) / (z2 - z1)]

so, if z1 = 1, z2 = 1+i, z3 = i, I get

f(z) = (z - 1) / (iz + 1) = T1

Next, hmm... if I could invert this transformation, I should then get what I want?
Does this make any sense?
I'm going to assume it does, for a moment. Then:

T1^(-1) = (w + 1) / (1 - iw) = T2


Ok, now, T2(0), T2(1), T2(infinity) do not give me the expected result.
Something is wrong. Where is the flaw?

Thank you!



EDIT: Actually, I think I may be onto something..

T2(0) = 1
T2(1) = 1+i

But I'm unable to work out T2(infinity). What do I do with this infinity?

for a mobius transformation:



if , then

That was the missing piece of the puzzle, thanks!!