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Thread: Complex Analysis (Q on Riemann Sphere)

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

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

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

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

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

    Is it possible to do this question without finding f(C) first?
    Last edited by shinn; Nov 7th 2010 at 09:25 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by shinn View Post
    Let $\displaystyle f(z) = \frac{1-i}{z-i}, z \neq i$

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

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

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

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

    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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