Removable singularity

File:Graph of x squared undefined at x equals 2.svg

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, as defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{sinc}(z) = \frac{\sin z}{z} }

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

${\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .$

Formally, if $U\subset \mathbb {C}$ is an open subset of the complex plane Template:Tmath, $a\in U$ a point of Template:Tmath, and $f:U\smallsetminus \{a\}\rightarrow \mathbb {C}$ is a holomorphic function, then $a$ is called a removable singularity for $f$ if there exists a holomorphic function $g:U\rightarrow \mathbb {C}$ which coincides with $f$ on Template:Tmath. We say $f$ is holomorphically extendable over $U$ if such a $g$ 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 $a$ is equivalent to it being analytic at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} (proof), i.e. having a power series representation. Define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(z) = \begin{cases} (z - a)^2 f(z) & z \ne a ,\\ 0 & z = a . \end{cases} }

Clearly, Template:Tmath is holomorphic on Template:Tmath, and there exists

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, .}

We have Template:Tmath and Template:Tmath; therefore

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, .}

Hence, where Template:Tmath, we have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \ldots \, .}

However,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(z) = c_2 + c_3 (z - a) + \cdots \, .}

is holomorphic on Template:Tmath, thus an extension of Template:Tmath.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} such that Template:Tmath. If so, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is called a pole of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and the smallest such Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is the order of Template:Tmath. So removable singularities are precisely the poles of order Template:Tmath. A meromorphic function blows up uniformly near its other poles.
2. If an isolated singularity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} maps every punctured open neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \smallsetminus \{a\}} 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 Archived 2012-12-20 at archive.today