thumb|right|200px|A graph of a [[parabola with a removable singularity at ]]

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

: <math> \text{sinc}(z) = \frac{\sin z}{z} </math>

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

: <math> \text{sinc}(z) = \frac{1}{z}\left(\sum_{k=0}^{\infty} \frac{(-1)^kz^{2k+1{(2k+1)!} \right) = \sum_{k=0}^{\infty} \frac{(-1)^kz^{2k{(2k+1)!} = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \frac{z^6}{7!} + \cdots. </math>

Formally, if <math>U \subset \mathbb C</math> is an open subset of the complex plane , <math>a \in U</math> a point of , and <math>f: U\smallsetminus \{a\} \rightarrow \mathbb C</math> is a holomorphic function, then <math>a</math> is called a removable singularity for <math>f</math> if there exists a holomorphic function <math>g: U \rightarrow \mathbb C</math> which coincides with <math>f</math> on . We say <math>f</math> is holomorphically extendable over <math>U</math> if such a <math>g</math> exists.

Riemann's theorem

Riemann's theorem on removable singularities is as follows:

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

: <math>

h(z) = \begin{cases}

(z - a)^2 f(z) & z \ne a ,\\

0 & z = a .

\end{cases}

</math>

Clearly, is holomorphic on , and there exists

: <math>h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0</math>

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

: <math>h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, .</math>

We have and ; therefore

: <math>h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, .</math>

Hence, where , we have:

: <math>f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \ldots \, .</math>

However,

: <math>g(z) = c_2 + c_3 (z - a) + \cdots \, .</math>

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. 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 . If so, <math>a</math> is called a pole of <math>f</math> and the smallest such <math>m</math> is the order of . So removable singularities are precisely the poles of order . A meromorphic function blows up uniformly near its other poles.
  2. If an isolated singularity <math>a</math> of <math>f</math> is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an <math>f</math> maps every punctured open neighborhood <math>U \smallsetminus \{a\}</math> to the entire complex plane, with the possible exception of at most one point.

See also

  • Analytic capacity
  • Removable discontinuity
  • Removable singular point at Encyclopedia of Mathematics