Misplaced Pages

Removable singularity

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Undefined point on a holomorphic function which can be made regular
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.
Find sources: "Removable singularity" – news · newspapers · books · scholar · JSTOR (July 2021) (Learn how and when to remove this message)
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}}}

has a singularity at z = 0. This singularity can be removed by defining 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}}} 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 .}

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

Riemann's theorem

Riemann's theorem on removable singularities is as follows:

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

  1. f {\displaystyle f} is holomorphically extendable over a {\displaystyle a} .
  2. f {\displaystyle f} is continuously extendable over a {\displaystyle a} .
  3. There exists a neighborhood of a {\displaystyle a} on which f {\displaystyle f} is bounded.
  4. lim z a ( z a ) f ( z ) = 0 {\displaystyle \lim _{z\to a}(z-a)f(z)=0} .

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} is equivalent to it being analytic at 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}}}

Clearly, h is holomorphic on D { 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}

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 \,.}

We have c0 = h(a) = 0 and c1 = 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 \,.}

Hence, where z 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 \,.}

However,

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

is holomorphic on D, thus an extension of f {\displaystyle 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 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} . If so, a {\displaystyle a} is called a pole of f {\displaystyle f} and the smallest such m {\displaystyle m} is the order of a {\displaystyle a} . 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 a {\displaystyle a} of 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} maps every punctured open neighborhood U { a } {\displaystyle U\setminus \{a\}} to the entire complex plane, with the possible exception of at most one point.

See also

External links

Categories:
Removable singularity Add topic