 Möbius transformation

A Möbius transformation of the Riemann sphere has the formThe group is generated by translations, nonzero complex scalings, and inversion $z\mapsto 1/z$.

A Möbius transformation (or linear fractional transformation) is a function defined on the complex numbers that has the general form: \[ f(z) = \frac{az + b}{cz + d} \] where \(a\), \(b\), \(c\), and \(d\) are complex numbers, and \(ad - bc \neq 0\) to ensure that the transformation is well-defined (i.e., it is not degenerate).