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Bilinear Maps Complex Analysis

  1. Sep 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Find a function [tex] g [/tex] analytic in [tex]|z|\leq 2[/tex], with [tex] g(2/3)=0[/tex] and [tex] |g(z)|= 1 [/tex] on [tex]|z|=2[/tex]

    2. Relevant equations
    Bilinear maps

    [tex] B_{\alpha}(z)=\frac{z-\alpha}{1-\overline{\alpha}z} [/tex]

    [tex] |B_{\alpha}(z)|=1[/tex] on [tex]|z|=1[/tex]

    3. The attempt at a solution
    I tried using the maximum modulus theorem but I did not manage to find such a function.
  2. jcsd
  3. Sep 10, 2009 #2


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    Science Advisor
    Homework Helper

    Why are you messing around with the maximum modulus theorem when your relevant equation is a bilinear map? Use the bilinear map. It maps the unit circle |z|=1 to the unit circle. Modify it so it maps the circle |z|=2 to |z|=1. Now you still have an 'a' in the function. Fix 'a' so that g(2/3)=0.
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