Bilinear Maps Complex Analysis

  • Thread starter Sistine
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  • #1
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Homework Statement


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]


Homework 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]


The Attempt at a Solution


I tried using the maximum modulus theorem but I did not manage to find such a function.
 

Answers and Replies

  • #2
Dick
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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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