# 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)