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Show the following (complex) function is analytic

  1. Apr 19, 2010 #1
    1. The problem statement, all variables and given/known data
    Given that f is analytic in the open disc of radius one around zero, and that f(0)=0 and mod(f(z))<=1 for ever z in D(0;1).

    Define g(z) = f'(0) if z=0, f(z)/z elsewhere.

    I want to show g is analytic in D(0;1)


    2. Relevant equations
    Basic stuff about analytic functions and series representations, etc.


    3. The attempt at a solution

    Okay, so if z does not equal 0 then it is the quotient of two analytic functions (f is analytic by the hypothesis and any polynomial is analytic) hence g is analytic.

    If z=0 then I wrote f as a Taylor series: f(z) = f(0) + f'(0)z + (f''(0)/2)*z^2 +...
    so g'(0) = [f'(0)]' = [0]' = 0 thus g is differentiable at z=0; and I'm done.

    Is my reasoning correct?
     
  2. jcsd
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