In mathematics, the Riemann sphere, named after Bernhard Riemann,^[1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ${\displaystyle \infty }$ for infinity. With the Riemann model, the point ${\displaystyle \infty }$ is near to very large numbers, just as the point ${\displaystyle 0}$ 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 ${\displaystyle 1/0=\infty }$ 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 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, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line ${\displaystyle \mathbf {P} ^{1}(\mathbf {C} )}$, the projective space of all complex lines in ${\displaystyle \mathbf {C} ^{2}}$. As with any compact 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 and in other branches of physics.

Extended complex numbers

The extended complex numbers consist of the complex numbers ${\displaystyle \mathbf {C} }$ together with ${\displaystyle \infty }$. The set of extended complex numbers may be written as ${\displaystyle \mathbf {C} \cup \{\infty \}}$, and is often denoted by adding some decoration to the letter ${\displaystyle \mathbf {C} }$, such as

${\displaystyle {\widehat {\mathbf {C} }},\quad {\overline {\mathbf {C} }},\quad {\text{or}}\quad \mathbf {C} _{\infty }.}$

The notation ${\displaystyle \mathbf {C} ^{*}}$ has also seen use, but as this notation is also used for the punctured plane ${\displaystyle \mathbf {C} \setminus \{0\}}$, it can lead to ambiguity.^[2]

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 ${\displaystyle z\in \mathbf {C} }$,

${\displaystyle z+\infty =\infty }$

for any complex number ${\displaystyle z}$, and multiplication may be defined by

${\displaystyle z\times \infty =\infty }$

for all nonzero complex numbers ${\displaystyle z}$, with ${\displaystyle \infty \times \infty =\infty }$. Note that ${\displaystyle \infty -\infty }$ and ${\displaystyle 0\times \infty }$ are left undefined. Unlike the complex numbers, the extended complex numbers do not form a field, since ${\displaystyle \infty }$ does not have an additive nor multiplicative inverse. Nonetheless, it is customary to define division on ${\displaystyle \mathbf {C} \cup \{\infty \}}$ by

${\displaystyle {\frac {z}{0}}=\infty \quad {\text{and}}\quad {\frac {z}{\infty }}=0}$

for all nonzero complex numbers ${\displaystyle z}$ with ${\displaystyle \infty /0=\infty }$ and ${\displaystyle 0/\infty =0}$. The quotients ${\displaystyle 0/0}$ and ${\displaystyle \infty /\infty }$ are left undefined.

Rational functions

Any rational function ${\displaystyle f(z)=g(z)/h(z)}$ (in other words, ${\displaystyle f(z)}$ is the ratio of polynomial functions ${\displaystyle g(z)}$ and ${\displaystyle h(z)}$ of ${\displaystyle z}$ with complex coefficients, such that ${\displaystyle g(z)}$ and ${\displaystyle h(z)}$ have no common factor) can be extended to a continuous function on the Riemann sphere. Specifically, if ${\displaystyle z_{0}}$ is a complex number such that the denominator ${\displaystyle h(z_{0})}$ is zero but the numerator ${\displaystyle g(z_{0})}$ is nonzero, then ${\displaystyle f(z_{0})}$ can be defined as ${\displaystyle \infty }$. Moreover, ${\displaystyle f(\infty )}$ can be defined as the limit of ${\displaystyle f(z)}$ as ${\displaystyle z\to \infty }$, which may be finite or infinite.

The set of complex rational functions—whose mathematical symbol is ${\displaystyle \mathbf {C} (z)}$—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 ${\displaystyle \infty }$ everywhere. The functions of ${\displaystyle \mathbf {C} (z)}$ form an algebraic field, known as the field of rational functions on the sphere.

For example, given the function

${\displaystyle f(z)={\frac {6z^{2}+1}{2z^{2}-50}}}$

we may define ${\displaystyle f(\pm 5)=\infty }$, since the denominator is zero at ${\displaystyle \pm 5}$, and ${\displaystyle f(\infty )=3}$ since $$Zdroj:https://en.wikipedia.org?pojem=Riemann_sphere
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