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Complex Analysis problem

  1. Jul 11, 2015 #1
    1. The problem statement, all variables and given/known data
    With Screen_Shot_2015_07_12_at_00_07_03.png . Give an example, if it exists, of a non constant holomorphic function Screen_Shot_2015_07_12_at_00_07_24.png that is zero everywhere and has the form 1/n, where n € N.
    2. Relevant equations
    So.. This was in my Complex Analysis exam, and i have no idea what to do. I always seem to get stuck at these more abstract questions.
     
  2. jcsd
  3. Jul 11, 2015 #2

    FactChecker

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    It seems like your problem statement must be wrong. How can a function be zero everywhere and nonzero at some points? Am I reading it wrong?
     
  4. Jul 11, 2015 #3

    WWGD

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    Just declare the function to be $0$ on a subset of the plane that has a limit point. But yes, the statement is kind of confusing.
     
  5. Jul 12, 2015 #4
    I tried to fix it.
    I'm so sorry, but i translated this. But even in my mother togue it's confusing.
     
  6. Jul 12, 2015 #5

    WWGD

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    The only way I can make sense of it is this: find an analytic function that has zeros at all points ##1/n, n=1,2,## and so that ##f(0)=0##. And then you use the identity theorem to conclude f must be identically ##0.##
     
    Last edited: Jul 12, 2015
  7. Jul 12, 2015 #6

    FactChecker

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    I bet they want a function that is zero at the points 1/n, analytic on the right half plane, and not constant.
     
  8. Jul 13, 2015 #7

    RUber

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    This must be some kind of a trick, right? So satisfy the conditions one by one, and then satisfy the "every point" condition by making it the limit of some sequence.
    1) Holomorphic - start thinking sine / cosine
    2) looks like 1/n at x = n, so maybe ##\frac{1}{x} ## times some sine or cosine function
    3) returns zero at every point, maybe the limit of your frequency component of the sine or cosine, so that at every point, you will have a vertical line from -1/x to 1/x.
     
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