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.