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Complex analysis - something really confusing

  1. Apr 7, 2006 #1
    I think I have misunderstood one of the theorems in complex analysis

    (k reperesents the order of the derivative)


    Theorem: Suppose f is analytic on a domain D and, further, at some point z0 subset of D, f (k) (z0) = 0. Then f(z) = 0 for all z subset of D ...

    Is the theorem basically saying is that if f(z) equals 0 at any z0, then it will equal zero for all of the points?? That doesnt' sound right at all ...

    any help with be greatly appreciated
     
  2. jcsd
  3. Apr 7, 2006 #2
    That theorem is obviously wrong. The function [tex]f(z) = z^{k+1}[/tex] is a counterexample. It verifies [tex]f^{k}(0) = 0[/tex], is holomorphic in any domain, yet it is nonzero for every [tex]z\ne 0[/tex].

    Most probably your theorem is one of the following:

    1) If [tex]f^k(z_0) = 0[/tex] for all [tex]k\ge 0[/tex], then f=0 in D.

    2) If [tex]f=0[/tex] in some open subset of D, then f=0 in D.
     
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