Conformal mappings between open disk and half space

Question

In which book / handout can I find an explicit description of a conformal mapping between the open ball and the upper half space in $n$ dimensions?

Answer 1

$\newcommand{\Brak}[1]{\langle#1\rangle}\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}$Fix $n \geq 2$, and write the general element of $\Reals^{n+1}$ as $$ (x_{1}, \dots, x_{n}, x_{n+1}) = (\Vec{x}, x_{n+1}). $$ Let $p:S^{n} \to \Reals^{n} \times \{0\} \subset \Reals^{n+1}$ denote stereographic projection from the "north pole" $\Vec{e}_{n+1} = (0, \dots, 0, 1)$, and $p^{-1}$ the inverse mapping. Finally, let $\Vec{u}$ be a unit vector in $\Reals^{n}$, and let $R$ denote the quarter-turn of $S^{n}$ in the plane containing $\Vec{u}$ and $\Vec{e}_{n+1} = (0, \dots, 0, 1)$ that sends $\Vec{u}$ to $\Vec{e}_{n+1}$. The composition $p \circ R \circ p^{-1}$ maps the open unit ball in $\Reals^{n}$ to the half-space through the origin with inward-pointing unit normal vector $\Vec{u}$.

Geometrically, $p^{-1}$ maps the open unit ball to the "open southern hemisphere" $\{x_{n+1} < 0\}$; the rotation $R$ sends the southern hemisphere to the hemisphere $H_{\Vec{u}}$ centered on $\Vec{u}$ (viewing $\Vec{u}$ as a point of $S^{n}$), which has the north pole on its boundary; and $p$ sends $H_{\Vec{u}}$ to the stated half space.

In coordinates, $$ p(\Vec{x}, x_{n+1}) = \frac{(\Vec{x}, 0)}{1 - x_{n+1}},\qquad p^{-1}(\Vec{x}, 0) = \frac{(2\Vec{x}, \|\Vec{x}\|^{2} - 1)}{\|\Vec{x}\|^{2} + 1}. $$ Every vector $(\Vec{x}, x_{n+1})$ in $\Reals^{n+1}$ may be decomposed as $$ (\Vec{x}, x_{n+1}) = \Brak{\Vec{x}, \Vec{u}} \Vec{u} + x_{n+1}\Vec{e}_{n+1} + \Vec{x}^{\perp}, $$ and $$ R(\Vec{x}, x_{n+1}) = \Brak{\Vec{x}, \Vec{u}} \Vec{e}_{n+1} - x_{n+1}\Vec{u} + \Vec{x}^{\perp}. $$ The composition is $$ p \circ R \circ p^{-1}(\Vec{x}, 0) = \frac{1}{\|\Vec{x}\|^{2} - 2\Brak{\Vec{x}, \Vec{u}} + 1}\bigl((1 - \|\Vec{x}\|^{2}) \Vec{u} + 2\Vec{x}^{\perp}, 0\bigr). $$

For example, if $\Vec{u} = \Vec{e}_{1} = (1, 0, \dots, 0)$, then \begin{align*} p \circ R \circ p^{-1}(\Vec{x}, 0) &= \frac{1}{\|\Vec{x}\|^{2} - 2x_{1} + 1}\bigl((1 - \|\Vec{x}\|^{2}), 2x_{2}, \dots, 2x_{n}, 0\bigr) \\ &= \frac{1}{(x_{1} - 1)^{2} + x_{2}^{2} + \dots + x_{n}^{2}}\bigl((1 - \|\Vec{x}\|^{2}), 2x_{2}, \dots, 2x_{n}, 0\bigr). \end{align*}