1. Mar 30, 2010 #1

   sbashrawi

   

   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

    

   sbashrawi, Mar 30, 2010
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3. Mar 31, 2010 #2

   Eynstone

   

   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.

    

   Eynstone, Mar 31, 2010
4. Mar 31, 2010 #3

   sbashrawi

   

   Thank you

    

   sbashrawi, Mar 31, 2010

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