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Math Help - Complex Analysis (Q on Riemann Sphere)

  1. #1
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    Complex Analysis (Q on Riemann Sphere)

    Let  f(z) = \frac{1-i}{z-i}, z \neq i and  \phi: \mathbb{C} \to \mathbb{S} be the stereographic projection of  \mathbb{C} into the Riemann Sphere \mathbb{S}.

    Explain why the image  \phi \circ f(C) in \mathbb{S} of the set

    C = \{ z \in \mathbb{C}: z \neq i, |z-1-i | =1\}

    by the map \phi \circ f lies on a circle through  \infty , the north pole of \mathbb{S}.

    Is it possible to do this question without finding f(C) first?
    Last edited by shinn; November 7th 2010 at 10:25 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by shinn View Post
    Let  f(z) = \frac{1-i}{z-i}, z \neq i

    (a) Let  \phi: \mathbb{C} \to \mathbb{S} be the stereographic projection of  \mathbb{C} into the Riemann Sphere \mathbb{S}.

    Explain why the image  \phi \circ f(C) in \mathbb{S} of the set

    C = \{ z \in \mathbb{C}: z \neq i, |z-1-i | =1\}

    by the map \phi \circ f lies on a circle through  \infty , the north pole of \mathbb{S}.

    Is it possible to do this question without finding f(C) first?
    Yes, and is this homework?
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  3. #3
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    No, its from a final past exam paper from UNSW (unversity of new south wales) for MATH2620, November 2007. I am just working through these questions preparing for my final exam and they don't have any solutions.
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