Encyclopedia > Pole (complex analysis)

Article Content

Pole (complex analysis)

In complex analysis, a pole of a function is a certain type of simple singularity which behaves like the singularity of f(z) = 1/zn at z = 0.

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 f(z) = g(z) / (z - a)n 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 limza f(z) exists as a complex number. If the order is bigger than 0, then limza f(z) = ∞.

A singularity which is not a pole is called an essential singularity.

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Bugatti ... Between 1958 and 1975 (when their business failed) they secretly amassed a remarkable collection of the cars. Now known as the Schlumpf Collection[?], it has been turned ...