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Alternate form of entire function

  1. Apr 25, 2012 #1
    1. The problem statement, all variables and given/known data
    f(z) entire, f(z) ≠ 1/z for z≠0. Show f(z) can be written as [1-exp(zh(z))]/z, h entire.


    2. Relevant equations
    None needed.


    3. The attempt at a solution
    After some algebra, you can rewrite it so that f(z) = [1-exp(g(z))]/z, where g is entire. So I get close.

    Because it is entire, we need to be careful at 0 to prevent a pole, so exp(g(0)) = 1 must be true. Therefore, g(0) = 2*pi*i*k, where k is an integer. Problem is, at that point I get stuck. The zh(z) looks a lot like the zf(z) in the initial entire function, but not sure how to turn g(z) into 2*pi*i*k + zh(z), h entire, and then the first term goes away any way because exp(2*pi*i*k) = 1.

    I'm sure it is some small statement that allows me to do that, but I'm overlooking it. Thanks!
     
  2. jcsd
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