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Wikipedia / Removable singularity

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.

For instance, the (unnormalized) sinc function, as defined by

has a singularity at . This singularity can be removed by defining , which is the limit of as tends to . The resulting function is holomorphic. In this case the problem was caused by being given an indeterminate form. Taking a power series expansion for around the singular point shows that

is an open subset of the complex plane a point of is a holomorphic function is called a removable singularity on exists.

Riemann's theorem

Riemann's theorem on removable singularities is as follows:

(proof), i.e. having a power series representation. Define

Clearly, is holomorphic on , and there exists

by 4, hence is holomorphic on and has a Taylor series about :

We have and ; therefore

Hence, where , we have:

However,

is holomorphic on , thus an extension of .

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. such that is called a pole is the order of . So removable singularities are precisely the poless of order . A meromorphic function blows up uniformly near its other poles.

  2. is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem to the entire complex plane, with the possible exception of at most one point.

See also