Möbius transformation

The extended complex plane \(\mathbb{C}\cup\{\infty\}\) adds a single point at infinity. There the transformation is extended by \(f(-d/c)=\infty\) and \(f(\infty)=a/c\) when \(c\neq0\), and by \(f(\infty)=\infty\) when \(c=0\). The condition \(ad-bc\neq0\) is equivalent to \(f\) being nonconstant, since \(f'(z)=\frac{ad-bc}{(cz+d)^2}\).

The composition of two Möbius transformations is another one, and its coefficient matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is the product of the matrices; the inverse corresponds to the inverse matrix, and every nonzero multiple of the matrix gives the same transformation. For example, the inverse of \(f(z)=\frac{2z+1}{z+1}\) is \(f^{-1}(w)=\frac{w-1}{2-w}\).

Every Möbius transformation is obtained by composing translations \(z+b\), rotations and dilations \(az\) with \(a\neq0\), and the inversion \(1/z\). That is why it sends circles and lines to circles or lines: inversion maps the line \(\operatorname{Re}z=1\) to a circle through the origin, and a circle containing the point that goes to \(\infty\) becomes a line.

Three distinct points and their three images determine a unique Möbius transformation, so there is always one taking any three given points to any other three. A classic example is \(w=\frac{z-i}{z+i}\), which maps the upper half-plane onto the unit disk, because \(|z-i|<|z+i|\) exactly when \(\operatorname{Im}z>0\).