0
$\begingroup$

What is the image of horizontal line through $i$ under the Möbius tranformation that interchanges $0$ and $1$, and maps $-1$ to $1+i$?

1 Answers 1

1

The horizontal line through $\,i\,$ is $\,y=i\,$ , and the Moebius Transf. you want is

$M(z):=-\frac{z-1}{iz+1}\Longrightarrow M(z=x+i)=-\frac{x-1+i}{ix}=i-\frac{i}{x}-\frac{1}{x}=-\frac{1}{x}+\frac{x-1}{x}i$

Of course, we must require $\,x=Re(z)\neq 0\,$, unless we work in the extended complex plane.

  • 0
    Well, you can see that as the point at infinity of the line...which, in fact, makes the line look more like a circle.;)2012-12-20