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Pole (complex analysis)
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In complex analysis, a mathematical discipline, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of at . 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.
Definition
Formally, suppose U is an open subset of the complex plane C, a is an element of U and f : U \ ? C is a function which is holomorphic over its domain.
Discussion
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Encyclopedia
In complex analysis, a mathematical discipline, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of at . 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.
Definition
Formally, suppose U is an open subset of the complex plane C, a is an element of U and f : U \ ? 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 for all z in U \
holds, then a is called a pole of f. The smallest such n 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
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:
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.
Examples
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.
See also
External links
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