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

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.