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Math Help - Nice Complex Analysis Problem

  1. #1
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    Nice Complex Analysis Problem

    As I said I was taking an exam more than a month ago on Complex Analyis. One of the optional problems was (if I remember correcty):

    "Find all complex rational functions which have modulos 1 on the unit circle"*.

    Note: I cannot help you with this problem it uses a theorem which I never learned.


    *)I was told that this problem was a favorite problem to give as an examination problem to graduate students in a certain event (which I forgot the name of).
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  2. #2
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    It sounds as if they are asking you to rediscover the theory of Blaschke products.
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    Quote Originally Posted by Opalg View Post
    It sounds as if they are asking you to rediscover the theory of Blaschke products.
    This is exactly the term a professor told me it is. Can you post a proof?

    And do you think this is a too hard of a question for an exam?
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  4. #4
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    Here's a bare outline of how the proof goes. The set G of all complex rational functions which have modulus 1 on the unit circle is a group (under pointwise multiplication). For each complex number a with |a|<1, the function f_a(z) = \frac{z-a}{1-\bar{a}z} is in G.

    Now suppose that f is in G. If f has no zeros in the unit ball then f must be constant. (That is the tricky part of the argument. According to the Wikipedia page that I linked to in my previous comment above, "this fact is a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|f(z)|)".)

    If f does have zeros on the unit ball, call them a_1, a_2, ..., a_n (counted according to their multiplicity). Then f/(f_{a_1}f_{a_2}\cdots f_{a_n}) is a function in G with no zeros in the unit ball and is therefore constant. Hence f = \lambda f_{a_1}f_{a_2}\cdots f_{a_n} for some number \lambda with |\lambda|=1.

    That's the structure theorem for functions in G. Whether or not it's fair to ask it as an exam question depends on the students' background, I guess. I would say it's a pretty tough assignment for any student who hasn't previously seen the result.
    Last edited by Opalg; October 13th 2007 at 02:13 AM.
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  5. #5
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    The quick outline of the solution to this problem was presented to me using Schwartz-Reflection Principle.
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