Pole (complex analysis)

In complex analysis, a pole is a type of singularity of a function of a complex variable. In the neighbourhood of a pole, the function behave like a negative power.

A function f has a pole of order k, where k is a positive integer, with (non-zero) residue r at a point a if the limit

$\lim _{z\rightarrow a}f(z)(z-a)^{k}=r.\,$.

The pole is an isolated singularity if there is a neighbourhood of a in which f is holomorphic except at a. In this case the function has a Laurent series in a neighbourhood of a, so that f is expressible as a power series

$f(z)=\sum _{n=-k}^{\infty }c_{n}(z-a)^{n},\,$

where the leading coefficient $c_{-k}=r$.

An isolated singularity may be either removable, a pole, or an essential singularity.