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Thread: Uniform Convergence

  1. #1
    Junior Member
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    Uniform Convergence

    I need some help in proving the following lemma

    Lemma . if $\displaystyle f_j $ are holomorphic functions on $\displaystyle U $ and $\displaystyle f_j \rightarrow $ uniformly on compact subsets of $\displaystyle U $, then any derivatives
    $\displaystyle (\frac{\partial}{\partial z_1})^l (\frac{\partial}{\partial z_2})^k f_j $
    converges to
    $\displaystyle (\frac{\partial}{\partial z_1})^l (\frac{\partial}{\partial z_2})^k f $
    Uniformly on compacts sets.

    Proof:
    i tried to fix $\displaystyle P \in U $ and choose $\displaystyle r > 0 $ such that $\displaystyle \overline{D}^2(P,r) \subseteq U$. i have some problem expressing $\displaystyle f_j on D^2(P,r) $ as a Cauchy integral of $\displaystyle f_j $ on $\displaystyle \partial D^2(P,r) $ and differentiating it under integral sign.
    I need some help please
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  2. #2
    Junior Member
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    Re: Uniform Convergence

    i need some help guys...please help me!
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  3. #3
    Super Member girdav's Avatar
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    Re: Uniform Convergence

    What are $\displaystyle z_1$ and $\displaystyle z_2$?
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