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

has a singularity at z = 0. This singularity can be removed by defining , which is the limit of  as z tends to 0. 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

Formally, if  is an open subset of the complex plane ,  a point of , and  is a holomorphic function, then  is called a removable singularity for  if there exists a holomorphic function  which coincides with  on . We say  is holomorphically extendable over  if such a  exists.

Riemann's theorem 

Riemann's theorem on removable singularities is as follows:

 Theorem. Let  be an open subset of the complex plane,  a point of  and  a holomorphic function defined on the set .  The following are equivalent:

  is holomorphically extendable over . 
  is continuously extendable over . 
 There exists a neighborhood of  on which  is bounded.
 .

The implications 1 ⇒ 2  ⇒ 3  ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at  is equivalent to it being analytic at  (proof), i.e. having a power series representation. Define

Clearly, h is holomorphic on D \ {a}, and there exists

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

We have c0 = h(a) = 0 and c1 = h(a) = 0; therefore

Hence, where z ≠ a, we have:

However,

is holomorphic on D, thus an extension of f.

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 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.

See also
 Analytic capacity
 Removable discontinuity

External links 
Removable singular point at Encyclopedia of Mathematics

Category:Analytic functions
Category:Meromorphic functions
Category:Bernhard Riemann