Laurent series theorem statement assume that $f$ be analytic with the annular region say $r<|z-z_0|<R$. However I am not able to visualize annular regions for the following functions in their Laurent series expansion around $z_0 = 0$.

$f(z) = \frac{\sin z}{z} = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} + \ldots$

$f(z) = \frac{\cos z}{z} = \frac{1}{z} - \frac{z}{2!} + \frac{z^3}{4!} + \ldots$.

What is $r$ and $R$ in above two cases. My intuition says that $r = 0$ and $R = \infty$. But then how they define annular region? How to draw and visualize annular regions for above two functions.

Thank you very much


Your intuition is right: $r=0$ and $R= \infty$. The annular region is $ \mathbb C \setminus \{0\}$


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