In my german textbook on complex analysis (K. Fritzsche: Grundkurs Funktionentheorie) in the context of Laurent Series there is the following theorem stated (translated by me, so all errors are due to myself)

$\mathbf{3.1.8}$ Theorem concerning the "Laurent Separation"

Let $f$ be a holomorphic function on the ring domain $K_{r,R}:=\{z\in\Bbb C:r<|z-z_0|<R\}$. Then, there are unique holomorphic functions $$f^+:D_R(z_0)\to\Bbb C~~\textit{ and }~~f^-:\Bbb C\setminus\overline{D_r(z_0)}\to\Bbb C$$ such that $$f^++f^-=f~\textit{ on }~K_{r,R}(z_0)~~\textit{ and }~~|f^-(z_0)|\to0~\textit{ for }~|z|\to\infty$$

Here $\overline M$ denotes the closure of $M$, i.e. the unification of $M$ with all of its limit points, and $D_r(z_0)$ denotes the disk centered at $z_0$ with radius $r$, i.e $D_r(z_0):=\{z\in\Bbb C:|z-z_0|<r\}$.

However, my question is simple: what happens to $z\in\overline{D_R(z_0)}$? with is it necessary to exclude the closure of $D_r(z_0)$ while discarding the closure of $D_R(z_0)$?

Thanks in advance!

  • $\begingroup$ Holomorphic functions are usually defined on open sets. Also with that definition, the intersection of the two domains is exactly the ring domain $K_{r, R}$. $\endgroup$ – Martin R May 14 at 11:17
  • $\begingroup$ @MartinR Oh...oh... Totally forgot about this. Thank you for the reminder. $\endgroup$ – mrtaurho May 14 at 11:18

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