Removable singularity

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A graph of a parabola with a removable singularity at x = 2

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

[math] \text{sinc}(z) = \frac{\sin z}{z} [/math]

has a singularity at z = 0. This singularity can be removed by defining [math]\text{sinc}(0) := 1[/math], which is the limit of [math]\text{sinc}[/math] as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by [math]\text{sinc}[/math] being given an indeterminate form. Taking a power series expansion for [math]\frac{\sin(z)}{z}[/math] 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]\mathbb C[/math], [math]a \in U[/math] a point of [math]U[/math], and [math]f: U\setminus \{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 [math]U\setminus \{a\}[/math]. 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:

Theorem. Let [math]D \subset \mathbb C[/math] be an open subset of the complex plane, [math]a \in D[/math] a point of [math]D[/math] and [math]f[/math] a holomorphic function defined on the set [math]D \setminus \{a\}[/math]. The following are equivalent:

  1. [math]f[/math] is holomorphically extendable over [math]a[/math].
  2. [math]f[/math] is continuously extendable over [math]a[/math].
  3. There exists a neighborhood of [math]a[/math] on which [math]f[/math] is bounded.
  4. [math]\lim_{z\to a}(z - a) f(z) = 0[/math].

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, h is holomorphic on D \ {a}, 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 h is holomorphic on D and has a Taylor series about a:

[math]h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .[/math]

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

[math]h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .[/math]

Hence, where z ≠ a, we have:

[math]f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \cdots \, .[/math]

However,

[math]g(z) = c_2 + c_3 (z - a) + \cdots \, .[/math]

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:

  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 [math]\lim_{z \rightarrow a}(z-a)^{m+1}f(z)=0[/math]. If so, [math]a[/math] is called a 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 poles of order 0. A holomorphic 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 \setminus \{a\}[/math] to the entire complex plane, with the possible exception of at most one point.

See also

External links






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