Let S = {(ξ; η; ζ ) ∈ Σ : ζ ≥ ζ₀g}, where 0 < ζ₀ < 1 and let T be the corresponding

set in C. Show that T is the exterior of a circle centered at 0.

Im really confused by this problem, can anyone please help me.

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- Sep 13th 2011, 11:57 AMhabsfan31Complex Analysis - Stereographic Projection
Let S = {(ξ; η; ζ ) ∈ Σ : ζ ≥ ζ₀g}, where 0 < ζ₀ < 1 and let T be the corresponding

set in C. Show that T is the exterior of a circle centered at 0.

Im really confused by this problem, can anyone please help me. - Sep 15th 2011, 01:01 AMFernandoRevillaRe: Complex Analysis - Stereographic Projection

I suppose the "$\displaystyle g$" appearing in $\displaystyle S$ is a typo. Using the well known equations of the Stereographic Projection we get $\displaystyle \zeta=\frac{x^2+y^2}{1+x^2+y^2}\geq \zeta_0 \Leftrightarrow \ldots \Leftrightarrow x^2+y^2\geq \frac{ \zeta_0}{1- \zeta_0}$ and $\displaystyle (0,0,1)\to \infty$ . That is, $\displaystyle T$ is the the complement in $\displaystyle \Sigma$ of an open disk centered at $\displaystyle (0,0)$ and radius $\displaystyle r=\sqrt{\zeta_0/(1-\zeta_0)}$ (the complement,**not**the topological exterior).