In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number such that . If so, is called a pole of and the smallest such is the order of . So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its other poles. If an isolated singularity of is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an maps every punctured open neighborhood to the entire complex plane, with the possible exception of at most one point.
In light of Riemann's theorem given a non-removable singularity, one might ask whether there exists a natural number! such that . If so, is called a pole of and the smallest such is the order of . So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its other poles. If an isolated singularity of is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an maps every punctured open neighborhood to the entire complex plane, with the possible exception of at most one point.