Definitions

Pole (complex analysis)

In complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity at z = 0. This means that, in particular, a pole of the function f(z) is a point z = a such that f(z) approaches infinity uniformly as z approaches a.

Definition

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 function which is holomorphic over its domain. If there exists a holomorphic function g : UC and a nonnegative integer n such that

$f\left(z\right) = frac\left\{g\left(z\right)\right\}\left\{\left(z-a\right)^n\right\}$
for all z in U − {a}, then a is called a pole of f. The smallest number n satisfying above condition is called the order of the pole. A pole of order 1 is called a simple pole. A pole of order 0 is a removable singularity.

From above several equivalent characterizations can be deduced:

If n is the order of pole a, then necessarily g(a) ≠ 0 for the function g in the above expression. So we can put

$f\left(z\right) = frac\left\{1\right\}\left\{h\left(z\right)\right\}$

for some h that is holomorphic in an open neighborhood of a and has a zero of order n at a. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions.

Also, by the holomorphy of g, f can be expressed as:

$f\left(z\right) = frac\left\{a_\left\{-n\right\}\right\}\left\{ \left(z - a\right)^n \right\} + cdots + frac\left\{a_\left\{-1\right\}\right\}\left\{ \left(z - a\right) \right\} + sum_\left\{k geq 0\right\} a_k \left(z - a\right)^k.$

This is a Laurent series with finite principal part. The holomorphic function ∑k≥0ak (z - a)k (on U) is called the regular part of f. So the point a is a pole of order n of f if and only if all the terms in the Laurent series expansion of f around a below degree −n vanishes and the term in degree −n is not zero.

Remarks

If the first derivative of a function f has a simple pole at a, then a is a branch point of f. (The converse need not be true).

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

A complex function which is holomorphic except for some isolated singularities and whose only singularities are poles is called meromorphic.