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 \(f\). If \(f(z)=\sum_{k\in\ZZ}c_k(z-z_0)^k\) is the Laurent expansion of \(f\) on \(D_r(z_0)\setminus\{z_0\}\), the order of \(f\) at \(z_0\) is \[ v_{z_0}(f)\coloneqq\inf\{k\in\ZZ:c_k\neq0\}\in\ZZ\cup\{-\infty,+\infty\}, \] with \(v_{z_0}(f)=+\infty\) when every \(c_k\) is \(0\).
Let \(z_0\) be an isolated singularity of \(f\). Then:
\(z_0\) is a removable singularity if and only if \(v_{z_0}(f)\ge0\);
\(z_0\) is a pole if and only if \(v_{z_0}(f)\in\ZZ\) and \(v_{z_0}(f)<0\), and then the order of the pole is \(-v_{z_0}(f)\);
\(z_0\) is an essential singularity if and only if \(v_{z_0}(f)=-\infty\).
If \(f=g\) near \(z_0\) with \(g\) holomorphic on a disc about \(z_0\), the Taylor series of \(g\) is a Laurent expansion of \(f\), so by uniqueness of Laurent coefficients \(c_k=0\) for \(k<0\). Conversely, if \(c_k=0\) for \(k<0\), the power series \(\sum_{k\ge0}c_k(z-z_0)^k\) converges on \(D_r(z_0)\) and agrees with \(f\) off \(z_0\).
This is the equivalence of a pole of order \(n\) with a Laurent expansion \(\sum_{k\ge-n}c_k(z-z_0)^k\), \(c_{-n}\neq0\), proved in D-AUD6K.
An essential singularity is one that is neither removable nor a pole; by (a) and (b) this happens exactly when \(c_k\neq0\) for infinitely many \(k<0\), that is, when \(v_{z_0}(f)=-\infty\).
The function \(\sin(1/z)\) has an essential singularity at \(0\). Its Laurent expansion on \(\CC\setminus\{0\}\) is \[ \sin(1/z)=\sum_{m\ge0}\frac{(-1)^m}{(2m+1)!}\,z^{-(2m+1)}, \] which has nonzero coefficients \(c_k\) for infinitely many \(k<0\), so \(v_0(\sin(1/z))=-\infty\).