What is the image of horizontal line through $i$ under the Möbius tranformation that interchanges $0$ and $1$, and maps $-1$ to $1+i$?
Image of horizontal line under Möbius Transformations
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general-topology
complex-analysis
1 Answers
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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.
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0Which shows the image is the line $y=x+1$. – 2012-12-20
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0Well, that was supposed to be the OP's own deduction... – 2012-12-20
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0DonAntonio What if we are on extended complex plane? – 2012-12-20
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0when x=0 then I get $-\inf +i -\inf $, what is this, what kind of point? – 2012-12-20
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0Well, 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