# What happens to $z\in\overline{D_R(z_0)}$?

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 $$K_{r,R}:=\{z\in\Bbb C:r<|z-z_0|. Then, there are unique holomorphic functions $$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|.

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 $$D_R(z_0)$$?

• Holomorphic functions are usually defined on open sets. Also with that definition, the intersection of the two domains is exactly the ring domain $K_{r, R}$. – Martin R May 14 at 11:17