View source for Extended complex plane

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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====
<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====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,   "Functions of one complex variable" , Springer  (1978)</TD></TR></table>