1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    Dick

    User Avatar
    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook