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Math Help - Complex Analysis - Stereographic Projection

  1. #1
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    Complex Analysis - Stereographic Projection

    Let S = {(ξ; η; ζ ) Σ : ζ ζg}, where 0 < ζ < 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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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Complex Analysis - Stereographic Projection

    Quote Originally Posted by habsfan31 View Post
    Let S = {(ξ; η; ζ ) Σ : ζ ζg}, where 0 < ζ < 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.

    I suppose the " g" appearing in S is a typo. Using the well known equations of the Stereographic Projection we get \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 (0,0,1)\to \infty . That is, T is the the complement in \Sigma of an open disk centered at (0,0) and radius r=\sqrt{\zeta_0/(1-\zeta_0)} (the complement, not the topological exterior).
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