Pole (complex analysis)

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In the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of  \frac{1}{z^n} at z = 0. This means that, in particular, a pole of the function f(z) is a point a such that f(z) approaches infinity as z approaches a.

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Definition

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 : UC and a positive integer n, such that for all z in U \ {a}

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 few authors allow the order of a pole to be zero, in which case a pole of order zero is either a regular point or a removable singularity. However, it is more usual to require the order of a pole to be positive.

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 vanish and the term in degree −n is not zero.

Pole at infinity

It can be defined for a complex function the notion of having a pole at the point at infinity. In this case U has to be a neighborhood of infinity. For example, the exterior of any closed ball. Now, for using the previous definition a meaning for g being holomorphic at \infty should be given and also for the notion of "having" a zero at infinity as za does at the finite point a. Instead a definition can be given starting from the definition at a finite point by "bringing" the point at infinity to a finite point. The map z\mapsto 1/z does that. Then, by definition, a function, f, holomorphic in a neighborhood of infinity has a pole at infinity if the function f(1 / z) (which will be holomorphic in a neighborhood of z = 0), has a pole at z = 0, the order of which will be taken as the order of the pole at infinity.

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