Editing Removable singularity (section)

== 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:

#In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number <math>m</math> such that <math>\lim_{z \rightarrow a}(z-a)^{m+1}f(z)=0</math>. If so, <math>a</math> is called a '''[[pole (complex analysis)|pole]]''' of <math>f</math> and the smallest such <math>m</math> is the '''order''' of <math>a</math>. So removable singularities are precisely the [[pole (complex analysis)|pole]]s of order 0. A holomorphic function blows up uniformly near its other poles.
#If an isolated singularity <math>a</math> of <math>f</math> is neither removable nor a pole, it is called an '''[[essential singularity]]'''.  The [[Picard Theorem|Great Picard Theorem]] shows that such an <math>f</math> maps every punctured open neighborhood <math>U \setminus \{a\}</math> to the entire complex plane, with the possible exception of at most one point.