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 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.
- 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.
Read more about this topic: Removable Singularity
Famous quotes containing the word kinds:
“There are two kinds of timidity—timidity of mind, and timidity of the nerves; physical timidity, and moral timidity. Each is independent of the other. The body may be frightened and quake while the mind remains calm and bold, and vice versë. This is the key to many eccentricities of conduct. When both kinds meet in the same man he will be good for nothing all his life.”
—Honoré De Balzac (1799–1850)