How Do Mobius Transformations Map Complex Domains?

[sgcbell2]

Hi, I am currently working on problem to do with conformal mappings and Mobius Transformations.

My problem is:
Find the Mobius Transformation which carries the points 1, i, 0 to the points ∞, 0, 1 (in precisely this order). Find the image of the domain { z : 0 < x < t} under this Mobius Transformation, where t > 0 is some fixed positive real number.

I have found this Mobius transformation to be (z-i)/i(z-1).

I thought that in order to find the image of the domain given above, I should input different values along the boundaries x=0 and x=t into this mobius transformations to see where they are mapped to. I have done this but I'm finding it hard to see the shape of the image.

Maybe I am doing something wrong?

 

[BvU]

Hello sg and welcome to PF.
I'm not really qualified re conformal mapping, but from what I know from complex numbers, I see the function you found transforms 1+0i to $\infty$, 0+i to 0 and 0 + 0i to 1+0i.

I see x+0i is mapped to 1/(1-x) + ix/(1-x) which satisfies Re - I am = 1 , so I would expect a line I am = Re - 1

 

[sgcbell2]

That makes a lot of sense! Thank you