Essential singularity
An isolated singularity that is neither removable nor a pole -- its principal part has infinitely many nonzero terms.
An isolated singularity z_0 of f is essential if it is neither removable nor a pole, which happens exactly when the principal part of f's Laurent series at z_0 has infinitely many nonzero terms. Near an essential singularity f behaves wildly rather than blowing up cleanly to \infty: f comes arbitrarily close to nearly every complex value infinitely often on any punctured disk around z_0, however small, a phenomenon explored later in this chapter.
Examples
f(z)=e^{1/z} has an essential singularity at z=0, since its Laurent series \sum_{n=0}^\infty \frac1{n!z^n} has infinitely many negative-power terms.
f(z)=\sin(1/z) has an essential singularity at z=0 for the same reason.