91Ó°ÊÓ

Prove the general formula for inversion in a circle \(C\) centered at \(z_{0}\) with radius \(r .\) In particular, show in this case that $$ i_{C}(z)=\frac{r^{2}}{\left(\overline{z-z_{0}}\right)}+z_{0}. $$

Find a transformation of \(\mathbb{C}^{+}\) that rotates points about \(2 i\) by an angle \(\pi / 4\). Show that this transformation has the form of a Möbius transformation.

Determine the inverse stereographic projection function \(\phi^{-1}: \mathbb{C}^{+} \rightarrow \mathbb{S}^{2} .\) In particular, show that for \(z=x+y i \neq \infty\), $$ \phi^{-1}(x, y)=\left(\frac{2 x}{x^{2}+y^{2}+1}, \frac{2 y}{x^{2}+y^{2}+1}, \frac{x^{2}+y^{2}-1}{x^{2}+y^{2}+1}\right). $$

Prove that inversion in the unit circle maps the circle \((x-a)^{2}+(y-b)^{2}=r^{2}\) to the circle $$ \left(x-\frac{a}{d}\right)^{2}+\left(y-\frac{b}{d}\right)^{2}=\left(\frac{r}{d}\right)^{2} $$ where \(d=a^{2}+b^{2}-r^{2},\) provided that \(d \neq 0\).

Find a formula for reflection about the vertical line \(x=k\).

Find a formula for reflection about the horizontal line \(y=k\).

Prove that the cross ratio of four distinct complex numbers is a real number if and only if the four points lie on the same cline.

Prove that any pair of nonintersecting clines in \(\mathbb{C}\) may be mapped by a Möbius transformation to concentric circles.