Half-planes, the upper half-disc, the first quadrant, sectors, and strips are mapped onto one another by Möbius transformations, powers, $\Log$, $e^{iz}$, the Joukowski map, and $\sin$.Suggest a slogan
Write \(\HH=\ts{\Im z>0}\) for the upper half-plane, \(\DD^+=\ts{\abs z<1,\ \Im z>0}\) for the upper half-disc, \(Q_1=\ts{\Re z>0,\ \Im z>0}\) for the first quadrant, \(S_\beta=\ts{re^{i\theta}\st r>0,\ 0<\theta<\beta}\) for the sector of opening \(\beta\), \(\Sigma=\ts{0<\Im z<\pi}\) for the horizontal strip, and \(\Log\) for the principal branch of the logarithm. Each of the following maps is a biholomorphism from the stated domain onto the stated target.
| Domain | Target | Map |
|---|---|---|
| \(\CC\) | \(\CC\) | translation \(z\mapsto z+h\), \(h\in\CC\) |
| \(\CC\) | \(\CC\) | dilation \(z\mapsto cz\), \(c>0\) |
| \(\CC\) | \(\CC\) | rotation \(z\mapsto e^{i\theta}z\), \(\theta\in\RR\) |
| \(S_{\pi/n}\), \(n\ge1\) | \(\HH\) | \(z\mapsto z^n\) (PR-PDYJC) |
| \(\DD^+\) | \(Q_1\) | \(z\mapsto\frac{1+z}{1-z}\) (PR-PW4Z6) |
| \(\HH\) | \(\Sigma\) | \(z\mapsto\Log z\) (PR-XCDL5) |
| \(\DD^+\) | \(\ts{\Re w<0,\ 0<\Im w<\pi}\) | \(z\mapsto\Log z\) (PR-PELLF) |
| \(\ts{0<\Re z<\pi,\ \Im z>0}\) | \(\DD^+\) | \(z\mapsto e^{iz}\) |
| \(\DD^+\) | \(\HH\) | \(z\mapsto-\frac12\big(z+\frac1z\big)\) (PR-OTMIR) |
| \(\ts{-\frac\pi2<\Re z<\frac\pi2,\ \Im z>0}\) | \(\HH\) | \(z\mapsto\sin z\) (PR-3LBLV) |