 Solution

For an isolated pole of order $N\ge 1$, the Laurent coefficients must satisfyThen 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.