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Holomorphic function on the unit disc

  1. May 1, 2012 #1
    Does there exist a holomorphic function f(z) on the unit disc and satisfies f(1/n) = f(-1/n) = 1/n^3 for every n in N?
     
  2. jcsd
  3. May 1, 2012 #2

    micromass

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    There does not even exist a continuous function that does this.
     
  4. May 1, 2012 #3
    How can we vigorously prove that? I am thinking of construct a function g such that g(1/n) = g(-1/n) = 1/n^2 and consider f/g to do it. However I am stuck and cannot go on
     
  5. May 1, 2012 #4

    micromass

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    What would f(0) be?
     
  6. May 1, 2012 #5
    Last time I checked, [itex]z \mapsto |z|[/itex] was continuous...

    iamqsqsqs, try looking at the zeros of f(z) - z^3. Do they form an isolated set of points?
     
  7. May 1, 2012 #6
    ....
     
  8. May 1, 2012 #7
    Sorry, brain fart. I meant to say [itex]z \mapsto |z|^3[/itex]
     
  9. May 1, 2012 #8

    Hehe...yes, I supposed so. Happens to me all the time. Your answer to look at the zeroes of [itex]\,\,f(z)-z^3\,\,[/itex] pretty much wraps this up, though.

    DonAntonio
     
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