Möbius transformation/Proofs

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Main article: Möbius transformation

[edit] Fixed Points

The article claims that for $c\ne 0$ , the two roots are

$\gamma = \frac{(a - d) \pm \sqrt{(a - d)^2 + 4 c b}}{2 c}$

of the quadratic equation

$c \gamma^2 - (a - d) \gamma - b = 0 \ ,$

which follows from the fixed point equation

$\gamma={{a\gamma +b}\over {c\gamma +d}}$

by multiplying both sides with the denominator $c γ + d$ and collecting equal powers of $γ$ . Note that the quadratic equation degenerates into a linear equation if $c = 0$ , this corresponds to the situation that one of the fixed points is the point at infinity. In this case the second fixed point is finite if $a-d \ne 0$ otherwise the point at infinity is a fixed point "with multiplicity two" (the case of a pure translation).