= Upper half-plane
{disambiguate=complex analysis}
{title2=$\mathbb H$}
{wiki=Upper_half-plane}
= Complex upper half-plane
{synonym}
= Upper half-plane
{synonym}
The <upper half-plane> is the <open set>
$$
\mathbb H=\{z=x+iy\in\mathbb C:y>0\}.
$$
It contains precisely the <complex numbers> with positive <imaginary part>: $1+i$ is inside, $1$ lies on its <boundary>, and $1-i$ belongs to the <lower half-plane>. In the <complex plane> its <boundary> is the <real line>; in the <Riemann sphere> the <boundary> also includes infinity. It is a <convex set>, hence a <simply connected domain>, and the <Cayley transform between the half-plane and disk> makes it <biholomorphic> to the <unit disc>.
With the <hyperbolic metric> $ds=|dz|/\operatorname{Im}z$, the same domain becomes the <Poincaré half-plane model>. Its <hyperbolic geodesics> are vertical lines and upper semicircles whose centres lie on the <real line>. Under the <Cayley transform>, these become arcs of <circles> orthogonal to the <unit circle>, with diameters as the straight-line cases. https://warwick.ac.uk/fac/sci/maths/people/staff/caroline_series/hyperbolic_geometry_ma448_lecture_notes.pdf[Series, Hyperbolic Geometry, sections 2.1 and 2.1.1]{ref}
\Image[/cayley-boundary-and-hyperbolic-routes.png]
{title=Boundary points and hyperbolic geodesics correspond under the Cayley transform from the upper half-plane to the unit disc}
Matching colours identify each <hyperbolic geodesic> and its image. The vertical line through zero maps to a diameter; the vertical line through one maps to an arc ending at $-i$ and $1$. A semicircle from $-1$ to $1$ also maps to a diameter, whereas the semicircle from $0$ to $2$ maps to a curved arc. Their endpoints lie on the <boundary>, outside the domains: the endpoints are infinitely far from interior points in <hyperbolic distance>.
The <upper half-plane> is useful because its straight <boundary> and explicit <upper half-plane automorphisms> make <conformal maps> and <hyperbolic geometry> calculable. The picture explains why shortest <hyperbolic geodesic> routes can look curved in Euclidean coordinates; their lengths are measured with the position-dependent <hyperbolic metric>.
For historical context, in his letter to Klein of 22 September 1882, Poincaré constructs a <quadrilateral> bounded by four <circles> orthogonal to a fundamental <circle>, and studies transformations generating repeated copies. In modern language, the sides inside that <circle> are <hyperbolic geodesics> in a disk model. The letter directly exhibits the connection between transformation <groups>, geometry and complex <functions>. https://henripoincarepapers.univ-nantes.fr/chp/text/klein-en-1882-09-22.html[Poincaré to Klein, 22 September 1882, English translation, opening construction]{ref}
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