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I'm stuck on a qualifying exam question, not really familiar with the method of solution. It goes as follows:

Let $f(z)$ be an an entire function and $g(z)$ be analytic in a neighborhood of $z=1$ which satisfies $g^{(n)}(1)=(f^{(n)}(1))^{\alpha}/(n!)^{\alpha-1}$, where $\alpha >0$. Show that $g(z)$ can be extended to an entire function.

Any help would be greatly appreciated.

edit: fixed typo

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    Yes, I apologize, that is correct. Just a typing a error on my part. $g^{(n)}(1)=$ etc.2012-08-17
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    I would guess that you can show that the taylor series for $g(z)$ is well-defined everywhere based on that condition.2012-08-17

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