# Extended complex plane

The complex $ z $-
plane $ \mathbf C $
compactified by adding the point $ \infty $
at infinity and written as $ \overline{\mathbf C}\; $.
The exterior of any circle in $ \mathbf C $,
that, is, any set of the form $ \{ \infty \} \cup \{ {z \in \mathbf C } : {| z - z _ {0} | > R } \} $,
$ R \geq 0 $,
becomes a neighbourhood of $ \infty $.
The extended complex plane is the Aleksandrov compactification of the plane $ \mathbf C $,
and is both homeomorphic and conformally equivalent to the Riemann sphere. The spherical, or chordal, metric on $ \overline{\mathbf C}\; $
is given by

$$ \rho ( z, w) = \ \frac{2 | z - w | }{\sqrt {1 + | z | ^ {2} } \sqrt {1 + | w | ^ {2} } } ,\ \ z, w \in \mathbf C , $$

$$ \rho ( z, \infty ) = { \frac{2}{\sqrt {1 + | z | ^ {2} } } } . $$

#### References

[1] | A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) (Translated from Russian) |

[2] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) |

#### Comments

#### References

[a1] | J.B. Conway, "Functions of one complex variable" , Springer (1978) |

**How to Cite This Entry:**

Extended complex plane.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Extended_complex_plane&oldid=46878