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 holomorphic function. If there exists a holomorphic function g : U > C and a natural number n such that f(z) = g(z) / (z  a)^{n} for all z in U  {a}, then a is called a pole of f. If n is chosen as small as possible, then n is called the order of the pole.
The number a is a pole of order n of f if and only if the Laurent series expansion of f around a has only finitely many negative degree terms, starting with (z  a)^{n}.
A pole of order 0 is a removable singularity. In this case the limit lim_{z→a} f(z) exists as a complex number. If the order is bigger than 0, then lim_{z→a} f(z) = ∞.
A singularity which is not a pole is called an essential singularity.
A holomorphic function whose only singularities are poles is called meromorphic.
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