Pole (complex analysis)

From Wikipedia, the free encyclopedia

The absolute value of the Gamma function. This shows that a function becomes infinite at the poles (left). On the right, the Gamma function does not have poles, it just increases quickly.
The absolute value of the Gamma function. This shows that a function becomes infinite at the poles (left). On the right, the Gamma function does not have poles, it just increases quickly.

In complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity 1/zn 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.

Contents

[edit] 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 : U → C and a nonnegative integer n such that

f(z) = \frac{g(z)}{(z-a)^n}

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(z) = \frac{1}{h(z)}

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

[edit] 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.

[edit] See also

[edit] External links