Construct Mobius Transform Easily: Step-by-Step Guide

[cheeez]

How do you construct mobius transform easily? is there a certain way to go about it or is it by inspection and experience. for ex. construct one from unit disc to left half plane or to right half plane, or below a certain y=ax+b line or something of the sort.

 

[cheeez]

any ideas

 

[lawsofform]

It is very easy. There is a theorem that says there is a mobius transformation to take any three points on the plane to any three other points. Typically, we take (a,b,c) to (0,1,$$\infty$$), because this transformation has the simple form:

$$f(z) = \frac{(z-a)(b-c)}{(z-c)(b-a)}$$

If you want to find a transformation that takes (a,b,c) to (a',b',c'), just find

$$f: (a,b,c)\rightarrow(0,1,\infty)$$
$$g: (a',b',c')\rightarrow(0,1,\infty)$$

then find the inverse of $$g$$ so

$$f \circ g^{-1}: (a,b,c)\rightarrow(a',b',c')$$ (LaTeX is not coming out right, this should read "f on g-inverse"

So say you want to take the unit disk to the right half plane. You want -i and i to go to points on the boundary such as 0 and $$\infty$$, and 0 go to a point in the right half plane such as 1. So

$$f(z) = \frac{(z+i)(0-i)}{(z-i)(0+i)}$$

You can check that this takes any point on the unit circle to the imaginary axis, and any point inside to the RHP.

Keep in mind that you can compose mobius transformations, so for example, to take the unit disk to the area under ax+b, you can find the transformation that takes the unit disk to the lower half plane, then find one to take the real line to the line ax+b.