I have encountered 2 similar but different theorems on expansions of holomorphic functions to power series, but am not sure how exactly do they differ.

Is correct that any $f: U \rightarrow \mathbb{C}$ holomorphic on $U$, for any closed ball $B(z_0, r)$ completely contained in $U$, $f$ identifies with a unique Taylor series on the ball?

And is it also correct that in the specific case that $U$ is a "ring" $\{z\space|\space r<|z|<R\}$, $f$ is representable on the entire ring by a unique Laurent series, but may not be representable on it by a single Taylor series?

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    $\begingroup$ Yes and yes. $\,\,\,$ $\endgroup$ – zhw. Jul 22 '16 at 17:24
  • $\begingroup$ (for a domain $r < |z| < R$ we say annulus : every Laurent series has an annlus as its domain of convergence) $\endgroup$ – reuns Jul 22 '16 at 17:25

I'm not quite sure what's going on in your first statement, with a set $U$ and closed balls inside $U$. All that matters is that $f$ be holomorphic on an open disk centered at the point in question - this is the disk on which the power series representation of $f$ will converge. Maybe you are worried about the kind of convergence? For both Taylor and Laurent series of holomorphic functions, the convergence is uniform on compact subsets of the disk/annulus.

Laurent series exist when $f$ is holomorphic on the annulus. It does not have to be holomorphic on the disk removed from the interior, and if I remember correctly, that removed disk could even be a point, i.e. $r=0$ is okay, and that $R = \infty$ is okay also.


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