Editing Möbius transformation (section)

=== Normal form ===
Möbius transformations are also sometimes written in terms of their fixed points in so-called '''normal form'''. We first treat the non-parabolic case, for which there are two distinct fixed points.

''Non-parabolic case'':

Every non-parabolic transformation is [[conjugacy class|conjugate]] to a dilation/rotation, i.e., a transformation of the form
<math display="block">z \mapsto k z </math>
{{nowrap|(''k'' ∈ '''C''')}} with fixed points at 0 and ∞. To see this define a map
<math display="block">g(z) = \frac{z - \gamma_1}{z - \gamma_2}</math>
which sends the points (''γ''<sub>1</sub>, ''γ''<sub>2</sub>) to (0, ∞). Here we assume that ''γ''<sub>1</sub> and ''γ''<sub>2</sub> are distinct and finite. If one of them is already at infinity then ''g'' can be modified so as to fix infinity and send the other point to 0.

If ''f'' has distinct fixed points (''γ''<sub>1</sub>, ''γ''<sub>2</sub>) then the transformation <math>gfg^{-1}</math> has fixed points at 0 and ∞ and is therefore a dilation: <math>gfg^{-1}(z) = kz</math>. The fixed point equation for the transformation ''f'' can then be written
<math display="block">\frac{f(z)-\gamma_1}{f(z)-\gamma_2} = k \frac{z-\gamma_1}{z-\gamma_2}.</math>

Solving for ''f'' gives (in matrix form):
<math display="block">\mathfrak{H}(k; \gamma_1, \gamma_2) =
\begin{pmatrix}
\gamma_1 - k\gamma_2 & (k - 1) \gamma_1\gamma_2 \\
1 - k                &  k\gamma_1 - \gamma_2
\end{pmatrix}</math>
or, if one of the fixed points is at infinity:
<math display="block">\mathfrak{H}(k; \gamma, \infty) =
\begin{pmatrix}
k & (1 - k) \gamma \\
0 &  1
\end{pmatrix}.</math>

From the above expressions one can calculate the derivatives of ''f'' at the fixed points:
<math display="block">f'(\gamma_1) = k </math> and <math display="block"> f'(\gamma_2) = 1/k.</math>

Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (''k'') of ''f'' as the '''characteristic constant''' of ''f''. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:
<math display="block">\mathfrak{H}(k; \gamma_1, \gamma_2) = \mathfrak{H}(1/k; \gamma_2, \gamma_1).</math>

For loxodromic transformations, whenever {{nowrap|{{abs|''k''}} > 1}}, one says that ''γ''<sub>1</sub> is the '''repulsive''' fixed point, and ''γ''<sub>2</sub> is the '''attractive''' fixed point. For {{nowrap|{{abs|''k''}} < 1}}, the roles are reversed.

''Parabolic case'':

In the parabolic case there is only one fixed point ''γ''. The transformation sending that point to ∞ is
<math display="block">g(z) = \frac{1}{z - \gamma}</math>
or the identity if ''γ'' is already at infinity. The transformation <math>gfg^{-1}</math> fixes infinity and is therefore a translation:
<math display="block">gfg^{-1}(z) = z + \beta\,.</math>

Here, ''β'' is called the '''translation length'''. The fixed point formula for a parabolic transformation is then
<math display="block">\frac{1}{f(z)-\gamma} = \frac{1}{z-\gamma} + \beta.</math>

Solving for ''f'' (in matrix form) gives
<math display="block">\mathfrak{H}(\beta; \gamma) =
\begin{pmatrix}
1+\gamma\beta    & - \beta \gamma^2  \\
\beta            &   1-\gamma \beta
\end{pmatrix}</math>Note that
<math>\det\mathfrak{H}(\beta;\gamma)=|\mathfrak{H}(\beta;\gamma) |
=\det
\begin{pmatrix}
1+\gamma\beta & -\beta\gamma^2\\
\beta & 1-\gamma\beta
\end{pmatrix}
=1-\gamma^2\beta^2+\gamma^2\beta^2=1

</math>

If {{nowrap|1=''γ'' = ∞}}:
<math display="block">\mathfrak{H}(\beta; \infty) =
\begin{pmatrix}
1 & \beta  \\
0 & 1
\end{pmatrix}</math>

Note that ''β'' is ''not'' the characteristic constant of ''f'', which is always 1 for a parabolic transformation. From the above expressions one can calculate:
<math display="block">f'(\gamma) = 1.</math>