Let \(z_0\in\CC\) and \(r>0\), and let \(f\) be holomorphic on the punctured disc \(D_r(z_0)\setminus\{z_0\}\). Then \(z_0\) is an isolated singularity of \(f\).
An isolated singularity \(z_0\) of \(f\) is of exactly one of three types, determined by the order \(v_{z_0}(f)\) of the Laurent expansion of \(f\) at \(z_0\):
a removable singularity, when \(v_{z_0}(f)\ge0\);
a pole, when \(v_{z_0}(f)\) is a negative integer;
an essential singularity, when \(v_{z_0}(f)=-\infty\).
The three conditions on \(v_{z_0}(f)\in\ZZ\cup\{\pm\infty\}\) are mutually exclusive and exhaustive, and each characterizes the corresponding type by the proposition on D-IWIA5.
The function \(\sin(1/z)\) has an essential singularity at \(0\): its Laurent expansion \(\sum_{m\ge0}\frac{(-1)^m}{(2m+1)!}z^{-(2m+1)}\) on \(\CC\setminus\{0\}\) has infinitely many nonzero coefficients of negative index.