Namespaces
Variants
Actions

Difference between revisions of "Extended complex plane"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
The complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369501.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369502.png" /> compactified by adding the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369503.png" /> at infinity and written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369504.png" />. The exterior of any circle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369505.png" />, that, is, any set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369507.png" />, becomes a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369508.png" />. The extended complex plane is the [[Aleksandrov compactification|Aleksandrov compactification]] of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e0369509.png" />, and is both homeomorphic and conformally equivalent to the [[Riemann sphere|Riemann sphere]]. The spherical, or chordal, metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e03695010.png" /> is given by
+
<!--
 +
e0369501.png
 +
$#A+1 = 12 n = 0
 +
$#C+1 = 12 : ~/encyclopedia/old_files/data/E036/E.0306950 Extended complex plane
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e03695011.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036950/e03695012.png" /></td> </tr></table>
+
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|Aleksandrov compactification]] of the plane  $  \mathbf C $,
 +
and is both homeomorphic and conformally equivalent to the [[Riemann sphere|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1–2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1–2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Functions of one complex variable" , Springer  (1978)</TD></TR></table>

Latest revision as of 19:38, 5 June 2020


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
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article