= Solution
For any circle $|\zeta|=\rho$ with $r<\rho<R$, Cauchy's formula applied inside and outside the circle gives the locally uniformly convergent Laurent expansion
$$
f(z)=\sum_{n=-\infty}^{\infty}a_nz^n,\qquad
a_n=\frac1{2\pi i}\oint_{|\zeta|=\rho}\frac{f(\zeta)}{\zeta^{n+1}}\,d\zeta.
$$
At zero, the singularity is removable exactly when all $a_n$ with $n<0$ vanish; it is a pole of order $k$ when $a_{-k}\ne0$ and $a_n=0$ for $n<-k$; it is essential when infinitely many negative coefficients are nonzero.
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