Pole

A pole of $k$ th-order is a isolated singularity such that in the Laurent series

f (z) = \sum_{n = 0}^{\infty} a_{n} (z - z_{0})^{n} + \sum_{n = 1}^{\infty} \frac{b_{n}}{(z - z_{0})^{n}}

we have $b_{k} \neq 0$ and $b_{n} = 0$ for $n > k$ . A removable singularity is a 0th-order pole.