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Successive derivative of analytic function

  1. Mar 30, 2010 #1
    1. The problem statement, all variables and given/known data
    Let f be analytic function on |z|<= R and lim fn(z)( the nth derivative) = f(z) uniformly in |z| <=R. Show that f(z) = c* exp(z)


    2. Relevant equations



    3. The attempt at a solution

    I tried the power series expansion but I couldn't prove it
     
  2. jcsd
  3. Mar 31, 2010 #2
    Let {Gn(z)} be a sequence of analytic functions converging to G(z), an analytic function. If the convergence is uniform in |z| <=R, then Gn'(z) converges to G'(z). (This can be proved either by an elementary limit argument or by Bach's differentiation formula).
    Here, Gn = fn -> f. As Gn' = fn'(z) = f_(n+1)(z), the same sequence converges to f' as well. Hence, f=f'. Taking h(z) =f(z) exp(-z), we find that h' =0 .QED.
     
  4. Mar 31, 2010 #3
    Thank you
     
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