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

${\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}}${\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}}

has a singularity at z = 0. This singularity can be removed by defining ${\displaystyle {\text{sinc}}(0):=1,}${\displaystyle {\text{sinc}}(0):=1,} which is the limit of sinc as z tends to 0. 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 ${\textstyle {\frac {\sin(z)}{z}}}${\textstyle {\frac {\sin(z)}{z}}} around the singular point shows that

${\displaystyle {\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .}${\displaystyle {\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 ${\displaystyle U\subset \mathbb {C} }${\displaystyle U\subset \mathbb {C} } is an open subset of the complex plane ${\displaystyle \mathbb {C} }${\displaystyle \mathbb {C} }, ${\displaystyle a\in U}${\displaystyle a\in U} a point of ${\displaystyle U}${\displaystyle U}, and ${\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} }${\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} } is a holomorphic function, then ${\displaystyle a}${\displaystyle a} is called a removable singularity for ${\displaystyle f}${\displaystyle f} if there exists a holomorphic function ${\displaystyle g:U\rightarrow \mathbb {C} }${\displaystyle g:U\rightarrow \mathbb {C} } which coincides with ${\displaystyle f}${\displaystyle f} on ${\displaystyle U\setminus \{a\}}${\displaystyle U\setminus \{a\}}. We say ${\displaystyle f}${\displaystyle f} is holomorphically extendable over ${\displaystyle U}${\displaystyle U} if such a ${\displaystyle g}${\displaystyle g} exists.

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 ${\displaystyle a}${\displaystyle a} is equivalent to it being analytic at ${\displaystyle a}${\displaystyle a} (proof), i.e. having a power series representation. Define

${\displaystyle h(z)={\begin{cases}(z-a)^{2}f(z)&z\neq a,\0円&z=a.\end{cases}}}${\displaystyle h(z)={\begin{cases}(z-a)^{2}f(z)&z\neq a,\0円&z=a.\end{cases}}}

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

${\displaystyle h'(a)=\lim _{z\to a}{\frac {(z-a)^{2}f(z)-0}{z-a}}=\lim _{z\to a}(z-a)f(z)=0}${\displaystyle h'(a)=\lim _{z\to a}{\frac {(z-a)^{2}f(z)-0}{z-a}}=\lim _{z\to a}(z-a)f(z)=0}

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

${\displaystyle h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots ,円.}${\displaystyle h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots ,円.}

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

${\displaystyle h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots ,円.}${\displaystyle h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots ,円.}

Hence, where ${\displaystyle z\neq a}${\displaystyle z\neq a}, we have:

${\displaystyle f(z)={\frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+\cdots ,円.}${\displaystyle f(z)={\frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+\cdots ,円.}

However,

${\displaystyle g(z)=c_{2}+c_{3}(z-a)+\cdots ,円.}${\displaystyle g(z)=c_{2}+c_{3}(z-a)+\cdots ,円.}

is holomorphic on D, thus an extension of ${\displaystyle f}${\displaystyle f}.

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 ${\displaystyle m}${\displaystyle m} such that ${\displaystyle \lim _{z\rightarrow a}(z-a)^{m+1}f(z)=0}${\displaystyle \lim _{z\rightarrow a}(z-a)^{m+1}f(z)=0}. If so, ${\displaystyle a}${\displaystyle a} is called a pole of ${\displaystyle f}${\displaystyle f} and the smallest such ${\displaystyle m}${\displaystyle m} is the order of ${\displaystyle a}${\displaystyle a}. So removable singularities are precisely the poles of order 0. A meromorphic function blows up uniformly near its other poles.
2. If an isolated singularity ${\displaystyle a}${\displaystyle a} of ${\displaystyle f}${\displaystyle f} is neither removable nor a pole, it is called an essential singularity . The Great Picard Theorem shows that such an ${\displaystyle f}${\displaystyle f} maps every punctured open neighborhood ${\displaystyle U\setminus \{a\}}${\displaystyle U\setminus \{a\}} to the entire complex plane, with the possible exception of at most one point.

• Analytic capacity
• Removable discontinuity

• Removable singular point at Encyclopedia of Mathematics Archived 2012年12月20日 at archive.today

• Analytic functions
• Meromorphic functions
• Bernhard Riemann

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