Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2004 / Analysis / 5 / b / Solution

A pole of order $m\ge 1$ means that $(z-a{)}^{m}f(z)$ extends holomorphically to $a$ and has a nonzero value there. Equivalently, the lowest nonzero power in the Laurent expansion is $(z-a{)}^{-m}$.