Complex numbers with non-negative imaginary part
In mathematics, the upper half-plane, is the set of points in the
Cartesian plane
with
The
lower half-plane is the set of points
with
instead. Each is an example of two-dimensional
half-space.
Affine geometry
The affine transformations of the upper half-plane include
- shifts , , and
- dilations ,
Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes
to .
- Proof: First shift the center of to Then take
and dilate. Then shift to the center of
Inversive geometry
Definition: .
can be recognized as the circle of radius centered at and as the
polar plot
of
Proposition: in and are
collinear points
.
In fact, is the
inversion
of the line
in the
unit circle. Indeed, the diagonal from
to
has squared length
, so that
is the reciprocal of that length.
Metric geometry
The distance between any two points and in the upper half-plane can be consistently defined as follows: The
perpendicular bisector
of the segment from
to
either intersects the boundary or is parallel to it. In the latter case
and
lie on a ray perpendicular to the boundary and
logarithmic measure
can be used to define a distance that is invariant under dilation. In the former case
and
lie on a circle centered at the intersection of their perpendicular bisector and the boundary. By the above proposition this circle can be moved by affine motion to
Distances on
can be defined using the correspondence with points on
and logarithmic measure on this ray. In consequence, the upper half-plane becomes a
.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the
imaginary part
:
The term arises from a common visualization of the complex number as the point in
Cartesian coordinates
. When the
axis is oriented vertically, the "upper
half-plane
" corresponds to the region above the
axis and thus complex numbers for which
.
It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by is equally good, but less used by convention. The
open unit disk
(the set of all complex numbers of
conformal mapping
to
(see "
Poincaré metric"), meaning that it is usually possible to pass between
and
It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.
The
.
The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.
Generalizations
One natural generalization in differential geometry is hyperbolic -space the maximally symmetric,
simply connected
,
-dimensional
Riemannian manifold with constant
sectional curvature . In this terminology, the upper half-plane is
since it has
In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product of copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space which is the domain of Siegel modular forms.
See also
References