Pole (complex analysis)
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 at z = 0. For a pole of the function f(z) at point a the function approaches infinity as z approaches a.
Formally, suppose U is an open subset of the complex plane C, p is an element of U and f : U \ {p} → C is a function which is holomorphic over its domain. If there exists a holomorphic function g : U → C, such that g(p) is nonzero, and a positive integer n, such that for all z in U \ {p}
holds, then p 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 p, then necessarily g(p) ≠ 0 for the function g in the above expression. So we can put
for some h that is holomorphic in an open neighborhood of p and has a zero of order n at p. So informally one might say that poles occur as reciprocals of zeros of holomorphic functions.
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