Editing Möbius transformation (section)

== Definition ==
The general form of a Möbius transformation is given by
<math display="block">f(z) = \frac{a z + b}{c z + d},</math>
where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are any [[complex number]]s that satisfy {{math|''ad'' − ''bc'' ≠ 0}}.

In case {{math|''c'' ≠ 0}}, this definition is extended to the whole [[Riemann sphere]] by defining
<math display="block">\begin{align}
f\left(\frac{-d}{c}\right) &= \infin, \\
f(\infin) &= \frac{a}{c}.\end{align}</math>

If {{math|1=''c'' = 0}}, we define
<math display="block">f(\infin) = \infin.</math>

Thus a Möbius transformation is always a bijective [[holomorphic function]] from the Riemann sphere to the Riemann sphere.

The set of all Möbius transformations forms a [[group (mathematics)|group]] under [[function composition|composition]]. This group can be given the structure of a [[complex manifold]] in such a way that composition and inversion are [[holomorphic map]]s. The Möbius group is then a [[complex Lie group]]. The Möbius group is usually denoted <math>\operatorname{Aut}(\widehat{\Complex})</math> as it is the [[automorphism group]] of the Riemann sphere.

If {{math|1=''ad'' = ''bc''}}, the rational function defined above is a constant (unless {{math|1=''c'' = ''d'' = 0}}, when it is undefined):
<math display="block">\frac{a z + b}{c z + d} = \frac{a}{c} = \frac{b}{d},</math>
where a fraction with a zero denominator is ignored.  A constant function is not bijective and is thus not considered a Möbius transformation.

An alternative definition is given as the kernel of the [[Schwarzian derivative]].