Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 5 / b

An isolated singularity at $z_0$ means that some radius $r>0$ has $f$ holomorphic on the punctured disk $0<|z-z_0|<r$, while $f$ is not initially defined as a holomorphic function at $z_0$. The adjective isolated refers to the absence of other singularities in this punctured neighbourhood; it permits a removable singularity.

The point is a removable singularity if assigning a value to $f(z_0)$ makes $f$ holomorphic on a full neighbourhood of $z_0$. Equivalently, its Laurent series has no nonzero coefficients of negative powers. The assigned value is its constant coefficient, or equivalently the finite limit ${\lim }_{z\to z_0}f(z)$.

A pole of order $n\ge 1$ meansEquivalently, the Laurent series has a finite nonzero principal part, with lowest power exactly $(z-z_0{)}^{-n}$. Thus $(z-z_0{)}^{n}f(z)$ extends holomorphically with nonzero value at $z_0$; using a smaller positive power would not remove the singularity.

An essential singularity is an isolated singularity that is neither a removable singularity nor a pole. In its Laurent series, infinitely many coefficients of negative powers are nonzero. The alternatives follow because no negative terms give a removable singularity, while finitely many with a lowest nonzero power give a pole.