1. Möbius transformation conformal

I would like to understand how to choose the right Mobius transformation. For example, the Möbius transformation that maps upper half-plane onto the unit disk is:

z --> (z-i)/(z+i)

The Möbius transformation that maps the unit disk one to one onto the right half-plane is:

z--> (1+z)/(1-z)

I would like to understand how one gets to this Möbius transformations. Thank you!

2. Re: Möbius transformation conformal

I'm not really sure what you're asking but maybe have a look at this

3. Re: Möbius transformation conformal

Choose three distinct points on the boundary of the upper half plane
move from left to right

e.g. $\{0,1,\infty \}$

choose three distinct points on the unit circle
moving counterclockwise

e.g. $\{-1,-i,1\}$

find a transformation that maps $\displaystyle 0$ to $\displaystyle -1$, $\displaystyle 1$ to $\displaystyle -i$, and $\displaystyle \infty$ to $\displaystyle 1$