 Isolated singularity

An isolated singularity of a complex function at $a$ is a missing or nonholomorphic point surrounded by a punctured disk $0<|z-a|<r$ on which the function is holomorphic. The punctured-disk hypothesis is decisive: it gives a Laurent series centered at $a$ and the three-way classification of isolated singularities.

At zero, $\sin z/z$ has a removable singularity, $z^{-2}$ has a double pole, and $e^{1/z}$ has an essential singularity. Their principal parts contain respectively no negative powers, finitely many negative powers, and infinitely many negative powers. For the first example, assigning the limiting value one fills the puncture holomorphically. For the second, multiplying by $z^2$ leaves a nonzero regular value. The third has the convergent punctured-plane expansion ${\sum }_{n\ge 0}{z}^{-n}/n!$.

An accumulation of other singularities changes the question. In reciprocal cosine with accumulating poles, every disk about zero contains poles, so zero is a non-isolated singularity and the same Laurent classification does not apply there. At infinity use the reciprocal coordinate $w=1/z$, as in pole at infinity and removable singularity at infinity.^[ref]

In complex analysis, an isolated singularity is a point at which a complex function is not defined or is not analytic, but is analytic in some neighborhood around that point, except at the singularity itself.