- #1

jimmycricket

- 116

- 2

## Homework Statement

Find the Mobius transformation which carries the points [tex]0,1,-i[/tex] to the points [tex]-1,0,\infty[/tex] respectively. Find the image of the domain [tex]\{z:x<0,-x+y<t\}[/tex] under this mobius transformation.

## Homework Equations

## The Attempt at a Solution

Let [tex]T(z)=\frac{az+b}{cz+d}[/tex].

Then [tex]T(0)=-1\Longrightarrow \frac{b}{d} \iff b=-d[/tex]

[tex]T(1)=0\Longrightarrow \frac{a+b}{c+d}=0\Longrightarrow a+b=0 \iff a=-b=d[/tex]

[tex]T(-i)=\infty\Longrightarrow \frac{-ia+b}{-ci+d}\iff d-ci=0\iff c=\frac{d}{i}=bi=-ai[/tex]

Now we have [tex]T(z)=\frac{az-a}{a-aiz}=\frac{z-1}{1-zi}[/tex]

So I now have to find the image under this map which is where I'm a bit stumped. Would it help to find where the intersections of the boundary of the domain with the axes are mapped to?