Removable singularity (original) (raw)

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Undefined point on a holomorphic function which can be made regular

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

sinc ( z ) = sin ⁡ z z {\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 sinc ( 0 ) := 1 , {\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 sin ⁡ ( z ) z {\textstyle {\frac {\sin(z)}{z}}} {\textstyle {\frac {\sin(z)}{z}}} around the singular point shows that

sinc ( z ) = 1 z ( ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! ) = ∑ k = 0 ∞ ( − 1 ) k z 2 k ( 2 k + 1 ) ! = 1 − z 2 3 ! + z 4 5 ! − z 6 7 ! + ⋯ . {\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 U ⊂ C {\displaystyle U\subset \mathbb {C} } {\displaystyle U\subset \mathbb {C} } is an open subset of the complex plane C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }, a ∈ U {\displaystyle a\in U} {\displaystyle a\in U} a point of U {\displaystyle U} {\displaystyle U}, and f : U ∖ { a } → C {\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} } {\displaystyle f:U\setminus \{a\}\rightarrow \mathbb {C} } is a holomorphic function, then a {\displaystyle a} {\displaystyle a} is called a removable singularity for f {\displaystyle f} {\displaystyle f} if there exists a holomorphic function g : U → C {\displaystyle g:U\rightarrow \mathbb {C} } {\displaystyle g:U\rightarrow \mathbb {C} } which coincides with f {\displaystyle f} {\displaystyle f} on U ∖ { a } {\displaystyle U\setminus \{a\}} {\displaystyle U\setminus \{a\}}. We say f {\displaystyle f} {\displaystyle f} is holomorphically extendable over U {\displaystyle U} {\displaystyle U} if such a g {\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 a {\displaystyle a} {\displaystyle a} is equivalent to it being analytic at a {\displaystyle a} {\displaystyle a} (proof), i.e. having a power series representation. Define

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

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

h ′ ( a ) = lim z → a ( z − a ) 2 f ( z ) − 0 z − a = lim z → 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} {\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:

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

We have _c_0 = h(a) = 0 and _c_1 = h'(a) = 0; therefore

h ( z ) = c 2 ( z − a ) 2 + c 3 ( z − a ) 3 + ⋯ . {\displaystyle h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots \,.} {\displaystyle h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots \,.}

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

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

However,

g ( z ) = c 2 + c 3 ( z − a ) + ⋯ . {\displaystyle g(z)=c_{2}+c_{3}(z-a)+\cdots \,.} {\displaystyle g(z)=c_{2}+c_{3}(z-a)+\cdots \,.}

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

Other kinds of singularities

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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 m {\displaystyle m} {\displaystyle m} such that lim z → a ( z − a ) m + 1 f ( z ) = 0 {\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, a {\displaystyle a} {\displaystyle a} is called a pole of f {\displaystyle f} {\displaystyle f} and the smallest such m {\displaystyle m} {\displaystyle m} is the order of a {\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 a {\displaystyle a} {\displaystyle a} of f {\displaystyle f} {\displaystyle f} is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an f {\displaystyle f} {\displaystyle f} maps every punctured open neighborhood U ∖ { a } {\displaystyle U\setminus \{a\}} {\displaystyle U\setminus \{a\}} to the entire complex plane, with the possible exception of at most one point.