Einheitskreis/Obere Halbebene/Rational isomorph/Reelle Koordinaten/Aufgabe/Lösung

Wir schreiben ${}w=a+b{\mathrm {i} }$ . Dann ist

{}{\begin{aligned}\psi (a+b{\mathrm {i} })&={\frac {{\left(a+b{\mathrm {i} }+1\right)}{\mathrm {i} }}{1-a-b{\mathrm {i} }}}\\&={\frac {a{\mathrm {i} }+{\mathrm {i} }-b}{1-a-b{\mathrm {i} }}}\\&={\frac {{\left(a{\mathrm {i} }+{\mathrm {i} }-b\right)}{\left(1-a+b{\mathrm {i} }\right)}}{{\left(1-a-b{\mathrm {i} }\right)}{\left(1-a+b{\mathrm {i} }\right)}}}\\&={\frac {{\left(a+1\right)}{\left(1-a\right)}{\mathrm {i} }-b^{2}{\mathrm {i} }-b{\left(1-a\right)}-b{\left(a+1\right)}}{{\left(1-a\right)}^{2}+b^{2}}}\\&={\frac {{\left(1-a^{2}-b^{2}\right)}{\mathrm {i} }-2b}{{\left(1-a\right)}^{2}+b^{2}}}.\end{aligned}}

In reellen Koordinaten ist daher

{}\psi {\begin{pmatrix}a\\b\end{pmatrix}}={\frac {1}{-2a+1+a^{2}+b^{2}}}{\begin{pmatrix}-2b\\1-a^{2}-b^{2}\end{pmatrix}}\,.