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Thread: sequence, analytic function

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    sequence, analytic function

    Let $\displaystyle \{ z_k \}$ be a sequence of distinct points in a domain $\displaystyle D$ that accumulates on $\displaystyle \partial D$. Let $\displaystyle \{ m_k \}$ be a sequence of positive integers, and for each $\displaystyle k$, let $\displaystyle a_{k0}, \ldots, a_{km_k}$ be complex numbers. Show that there is an analytic function $\displaystyle f(z)$ on $\displaystyle D$ such that $\displaystyle f^{(j)}(z_k)=a_{kj}$ for $\displaystyle 0 \leq j \leq m_k$ and for all $\displaystyle k$.

    In this section we have covered Runge's Theorem. However, I am still not sure how to prove this. I need some hints on doing this. Thanks in advance.
    Last edited by eskimo343; Apr 19th 2010 at 12:40 PM.
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