In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Riemann sphere, named after
Bernhard Riemann, is a
model of the extended complex plane: the
complex plane plus one
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
. This extended plane represents the extended complex numbers, that is, the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s plus a value
for
infinity. With the Riemann model, the point
is near to very large numbers, just as the point
is near to very small numbers.
The extended complex numbers are useful in
complex analysis because they allow for
division by zero in some circumstances, in a way that makes expressions such as
well-behaved. For example, any
rational function on the complex plane can be extended to a
holomorphic function on the Riemann sphere, with the
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in ...
of the rational function mapping to infinity. More generally, any
meromorphic function can be thought of as a holomorphic function whose
codomain is the Riemann sphere.
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Riemann sphere is the prototypical example of a
Riemann surface, and is one of the simplest
complex manifolds. In
projective geometry, the sphere can be thought of as the complex
projective line , the
projective space of all
complex lines in
. As with any
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
Riemann surface, the sphere may also be viewed as a projective
algebraic curve, making it a fundamental example in
algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the
Bloch sphere of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and in other
branches of physics.
The extended complex plane is also called the closed complex plane.
Extended complex numbers
The extended complex numbers consist of the complex numbers
together with
. The set of extended complex numbers may be written as
, and is often denoted by adding some decoration to the letter
, such as
:
The notation
has also seen use, but as this notation is also used for the punctured plane
, it can lead to ambiguity.
Geometrically, the set of extended complex numbers is referred to as the Riemann sphere (or extended complex plane).
Arithmetic operations
Addition of complex numbers may be extended by defining, for
,
:
for any complex number
, and
multiplication may be defined by
:
for all nonzero complex numbers
, with
. Note that
and
are left
undefined. Unlike the complex numbers, the extended complex numbers do not form a
field, since
does not have an
additive nor
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
. Nonetheless, it is customary to define
division on
by
:
for all nonzero complex numbers
with
and
. The quotients
and
are left undefined.
Rational functions
Any
rational function (in other words,
is the ratio of polynomial functions
and
of
with complex coefficients, such that
and
have no common factor) can be extended to a
continuous function on the Riemann sphere. Specifically, if
is a complex number such that the denominator
is zero but the numerator
is nonzero, then
can be defined as
. Moreover,
can be defined as the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of
as
, which may be finite or infinite.
The set of complex rational functions—whose mathematical symbol is
—form all possible
holomorphic functions from the Riemann sphere to itself, when it is viewed as a
Riemann surface, except for the constant function taking the value
everywhere. The functions of
form an algebraic field, known as ''the field of rational functions on the sphere''.
For example, given the function
:
we may define
, since the denominator is zero at
, and
since
as
. Using these definitions,
becomes a continuous function from the Riemann sphere to itself.
As a complex manifold
As a one-dimensional complex manifold, the Riemann sphere can be described by two charts, both with domain equal to the complex number plane
. Let
be a complex number in one copy of
, and let
be a complex number in another copy of
. Identify each nonzero complex number
of the first
with the nonzero complex number
of the second
. Then the map
:
is called the
transition map between the two copies of
—the so-called
charts—glueing them together. Since the transition maps are
holomorphic, they define a complex manifold, called the Riemann sphere. As a complex manifold of 1 complex dimension (i.e. 2 real dimensions), this is also called a Riemann surface.
Intuitively, the transition maps indicate how to glue two planes together to form the Riemann sphere. The planes are glued in an "inside-out" manner, so that they overlap almost everywhere, with each plane contributing just one point (its origin) missing from the other plane. In other words, (almost) every point in the Riemann sphere has both a
value and a
value, and the two values are related by
. The point where
should then have
-value "
"; in this sense, the origin of the
-chart plays the role of
in the
-chart. Symmetrically, the origin of the
-chart plays the role of
in the
-chart.
Topologically, the resulting space is the
one-point compactification of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined
complex structure, so that around every point on the sphere there is a neighborhood that can be
biholomorphically identified with
.
On the other hand, the
uniformization theorem, a central result in the classification of Riemann surfaces, states that every
simply-connected Riemann surface is biholomorphic to the complex plane, the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
, or the Riemann sphere. Of these, the Riemann sphere is the only one that is a
closed surface (a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
surface without
boundary). Hence the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.
As the complex projective line
The Riemann sphere can also be defined as the complex projective line. The points of the complex projective line are
equivalence classes established by the following relation on points from
: If for some
,
and
, then
.
In this case, the equivalence class is written
using
projective coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinate system, Cartesian coordinates are u ...
. Given any point
in the complex projective line, one of
and
must be non-zero, say
. Then by the equivalence relation,