Strictly more holomorphic functions on annulus than on punctured disc

Question

Consider the punctured disc $D=\{z\in\mathbb{C}:0<|z|<R\}$ and the annulus $A=\{z\in\mathbb{C}:r<|z|<R\}$. It is clear that every function holomorphic on $D$ is also holomorphic on $A$. But I need to show that there are strictly more functions holomorphic on $A$ than functions holomorphic on $D$. So I will have to find an example of a function holomorphic on $A$ that is not holomorphic on $D$. Could somebody give me an example? Does this have something to do with the Laurent series? Thanks for your help!

Answer 1

For example, $f(z)={1\over z-{r\over 2}}$.