Let \(f\) be holomorphic on a punctured disc \(D_r(z_0)\setminus\{z_0\}\). The isolated singularity \(z_0\) is an essential singularity of \(f\) if it is neither a removable singularity nor a pole of \(f\).
Let \(f(z)=\sum_{k\in\ZZ}c_k(z-z_0)^k\) be the Laurent expansion of \(f\) on \(D_r(z_0)\setminus\{z_0\}\). Then \(z_0\) is an essential singularity of \(f\) if and only if \(c_k\neq0\) for infinitely many \(k<0\), that is, if and only if the principal part \(\sum_{k<0}c_k(z-z_0)^k\) has infinitely many nonzero terms.
By the proposition on D-IWIA5, \(z_0\) is removable exactly when \(c_k=0\) for all \(k<0\), and a pole exactly when \(c_k\neq0\) for some but only finitely many \(k<0\). The remaining case is that \(c_k\neq0\) for infinitely many \(k<0\).