Möbius transformation

Definition

A Möbius transformation is a map of the Riemann sphere

T(z)=\frac{az+b}{cz+d}, \qquad a,b,c,d\in\mathbb C,\qquad ad-bc\ne0,

extended by $T(-d/c)=\infty$ and $T(\infty)=a/c$ when $c\ne0$ , with the evident affine conventions when $c=0$ .

Matrix origin

The matrix

A=\begin{pmatrix}a&b\\c&d\end{pmatrix}

acts linearly on $\mathbb C^2$ and hence projectively on its lines. Multiplying $A$ by a nonzero scalar does not change $T$ , so Möbius transformations are elements of $PGL_2(\mathbb C)$ . Composition corresponds to matrix multiplication.

Geometry

Every Möbius transformation is a biholomorphism of the sphere and carries generalized circles to generalized circles. It is conformal and orientation preserving. Reflection or complex conjugation produces anti-Möbius transformations, which are not holomorphic.

Generators

Translations, nonzero complex dilations, and inversion $z\mapsto1/z$ generate all Möbius transformations.

References

Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 3, §3.