Pole (complex analysis)

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

File:Gamma abs.png
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.

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 : UC 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. A pole of order 1 is called a simple pole.

Equivalently, a is a pole of order n≥ 0 for a function f if there exists an open neighbourhood U of a such that f : U - {a} → C is holomorphic and the limit

exists and is different from 0.

The point a is a pole of order n of f if and only if all the terms the Laurent series expansion of f around a below degree −n vanishes and the term in degree −n is not zero.

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

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 holomorphic function whose only singularities are poles is called meromorphic.

See also

de:Polstelle fr:Pôle (mathématiques) he:קוטב (אנליזה מרוכבת) it:polo (analisi complessa) pl:Biegun (matematyka) sl:pol (kompleksna analiza)