Find Mobius Transformations (M→N)

[tazzzdo]

Homework Statement

H is the upper-half plane model of the hyperbolic space

Find all Mobius transformations that send M to N.

Homework Equations

a) M = {0, 1, ∞}, N = {∞, 0, 1}
b) M = {0, 1, ∞}, N = {0, ∞, 2}
c) M = {i, -i, 3i}, N = {∞, i + 1, 6}

The Attempt at a Solution

Using the transformation:

Could I develop two equations m(z) and n(z) that map M and N to {0, 1, ∞}, respectively? Then just find n^-1\circm? Wouldn't that result in M→N?

If that's true, I'm just confused as to what points to use for z₁, z₂, z₃.

 

[Dick]

That works. Your transformation f1(z) takes z1->0, z2->1 and z3->∞. Which case of finding a specific transformation m(z) or n(z) are you confused about? Are you supposed to map the sets in the given order or in any order?

 

[tazzzdo]

Well if I plug those z's in, I get z(-∞)/(z-∞) for m(z). I seem to hit a wall there. Same with n(z). I don't know how to fix those equations into a form that I can use for the composition.

 

[Dick]

Well if I plug those z's in, I get z(-∞)/(z-∞) for m(z). I seem to hit a wall there. Same with n(z). I don't know how to fix those equations into a form that I can use for the composition.

Which part are you working on? I suspect it's trying to map {0,1,∞} to {0,1,∞}. If z is finite then (-∞)/(z-∞) should cancel to 1. That would mean m(z)=z should work. It does, doesn't it?

 

[tazzzdo]

Yeah I see what you're saying. Then does that mean I just switch the z's around in (a) for n(z)? But use the same transformation?

 

[tazzzdo]

I can't find a transformation n such that n(1) = infinity

 

[Dick]

I can't find a transformation n such that n(1) = infinity

Why not? Find it the same way you found m(z). What do you get when you put the z1, z2, z3 in?