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I've been working on some problems involving series and have found myself applying the identity principle to show that two representations of a series will lead them being unique under some conditions.

However in a more general setting, I haven't been able to answer this question:

Suppose I have a complex series defined on $D(0, 2)$ by $$f(z)= \sum a_n z^n$$ and I have another series defined on $D(3,3)$ by $$g(z)= \sum b_n (z-3)^n$$

If $g \equiv f$ on $D(0,2) \cap D(3,3)$ must all the coefficients be the same? Note that I left out the indices because I am not sure if the conclusion is different between taylor or laurent series.

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    $\begingroup$ what does it mean coefficients be the same? the power series are centered at different points and $3$ is not in the given domain of convergence for $f$ nor $0$ is in the given domain for $g$, so unclear what the question means; if for example, $g$ would converge on $D(3,4)$ say so $0$ is in there, then expanding $\sum b_n(z-3)^n$ in terms of $z^k$ and rearranging (which is permitted in this case) leads indeed to $a_n$ $\endgroup$
    – Conrad
    Commented May 20 at 19:54

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