Pole (complex analysis)

In complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f(z) = 1/zⁿ at z = 0; a pole of a function f is a point a such that f(z) approaches infinity as z approaches a.

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

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) = ∞.

The first derivative of f has a pole of order 1 (a "simple" pole) at a iff a is a branch point of f.

A non-removable singularity that is not a pole or a branch point is called an essential singularity.

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

See also

Zero (complex analysis)