= Solution
For an isolated <pole> of order $N\geq1$, the <Laurent coefficients> must satisfy
$$
\boxed{a_{-N}\ne0,\qquad a_n=0\quad(n<-N).}
$$
Then the <principal part of a Laurent series> is a finite sum with lowest power $-N$. The classification at the center assumes analyticity on a punctured neighborhood $0<|z-z_0|<b$, or an analytic continuation there. A Laurent expansion on an annulus with positive inner radius alone does not imply that the center is an isolated singularity.
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