Removable Singularity - Other Kinds of Singularities

Other Kinds of Singularities

Unlike functions of a real variable, holomorphic functions are sufficiently rigid that their isolated singularities can be completely classified. A holomorphic function's singularity is either not really a singularity at all, i.e. a removable singularity, or one of the following two types:

1. 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 poles.
2. If an isolated singularity of is neither removable nor a pole, it is called an essential singularity. It can be shown that such an maps every punctured open neighborhood to the entire complex plane, with the possible exception of at most one point.