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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.
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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

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

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≥0}a_{k} (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.

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.

- Zero (complex analysis)
- Residue (complex analysis)
- Electronic filter
- Control theory#Stability
- Pole–zero plot

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Last updated on Saturday October 11, 2008 at 07:15:58 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 07:15:58 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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