Möbius transformation article proofs

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This mathematics article is devoted entirely to providing proofs and backup support for claims and statements made in the article Möbius transformation. This article is currently an experimental vehicle to see how we might be able to provide proofs and details for math articles without cluttering up the main article itself. See Wikipedia:WikiProject Mathematics/Proofs for current discussion. This article is "experimental" in that its a proposal for one way that we might be able to deal with this.

[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 cγ + 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).

Note that

(a - d)^2 + 4 c b =(a - d)^2 + 4ad -4 = (a+d)^2-4 = \mbox{tr}^2\mathfrak{H} - 4


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