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 : U → C 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 (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.
Read more about this topic: Pole (complex Analysis)
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