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If a Möbius transform is a map that goes from the extended complex plane to the extended complex plane, given by some $\omega = \frac{az + b}{cz + d}$, where $ad - bc \neq 0$. In my notes, underneath the definition, I have:

Ex: $ \frac{1z + 2}{2z + 4} = \frac{1}{2}$.

How does this work? Isn't ad - bc = 0? Or have I made some kind of mistake in writing down the notes properly? Where does the fraction come from?

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    $2z+4=2(1z+2)$, so $\frac{1z+2}{2z+4}=\frac{1}{2}$2012-12-13

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As per your definition, the map is not a Möbius transform. As for how it's $1/2$? $2z+4=2(1z+2)$.