Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    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
    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.

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