Pole (complex analysis)

In complex analysis, a pole of a function is a certain type of simple singularity which behaves like the singularity of f(z) = 1/zⁿ at z = 0.

Formally, suppose U is an open subset of the complex plane C, a is an element of U and f : U - {a} -> C is a holomorphic function. If there exists a holomorphic function g : U -> C and a natural number n such that f(z) = g(z) / (z - a)ⁿ for all z in U - {a}, then a is called a pole of f. If n is chosen as small as possible, then n is called the order of the pole.

The number a is a pole of order n of f if and only if the Laurent series expansion of f around a has only finitely many negative degree terms, starting with (z - a)^-n.

A pole of order 0 is a removable singularity. In this case the limit lim_z→a f(z) exists as a complex number. If the order is bigger than 0, then lim_z→a f(z) = ∞.

A singularity which is not a pole is called an essential singularity.

A holomorphic function whose only singularities are poles is called meromorphic.