# Pole (complex analysis)

In complex analysis, a **pole** of a holomorphic function is a certain type of simple singularity that behaves like the singularity 1/*z*^{n} 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*.

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**. 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 lim_{z→a} *f*(*z*) exists as a complex number. If the order is bigger than 0, then lim_{z→a} *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)