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[[File:Graph x squared undefined at x equals two.png|thumb|right|200px|A graph of a [[parabola]] with a '''removable singularity''' at ''x'' = 2]] | |||
In [[complex analysis]], a '''removable singularity''' (sometimes called a '''cosmetic singularity''') of a [[holomorphic function]] is a point at which the function is undefined, but it is possible to define the function at that point in such a way that the function is regular in a neighbourhood of that point. | |||
For instance, the function | |||
:<math> f(z) = \frac{\sin z}{z} </math> | |||
has a singularity at ''z'' = 0. This singularity can be removed by defining ''f''(0) := 1, which is the [[Limit of a function|limit]] of ''f'' as ''z'' tends to 0. The resulting function is holomorphic. In this case the problem was caused by ''f'' being given an [[indeterminate form]]. Taking a power series expansion for <math>\frac{\sin(z)}{z}</math> shows that | |||
:<math> f(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 == | |||
[[Bernhard Riemann|Riemann's]] theorem on removable singularities states when a singularity is removable: | |||
''' Theorem.''' Let <math>D \subset 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: | |||
# <math>f</math> is holomorphically extendable over <math>a</math>. | |||
# <math>f</math> is continuously extendable over <math>a</math>. | |||
# There exists a [[neighborhood (topology)|neighborhood]] of <math>a</math> on which <math>f</math> is [[bounded function|bounded]]. | |||
# <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 that holomorphic functions are analytic|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 ''c''<sub>0</sub> = ''h''(''a'') = 0 and ''c''<sub>1</sub> = ''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)=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: | |||
#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 (complex analysis)|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 [[pole (complex analysis)|pole]]s of order 0. A holomorphic function blows up uniformly near its poles. | |||
#If an isolated singularity <math>a</math> of <math>f</math> is neither removable nor a pole, it is called an '''[[essential singularity]]'''. It can be shown 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== | |||
* [[Analytic capacity]] | |||
* [[Removable discontinuity]] | |||
== External links == | |||
{{Expand section|date=December 2009}} | |||
[[Category:Analytic functions]] | |||
[[Category:Meromorphic functions]] |
Revision as of 11:00, 30 January 2014
![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Graph_x_squared_undefined_at_x_equals_two.png/200px-Graph_x_squared_undefined_at_x_equals_two.png)
In complex analysis, a removable singularity (sometimes called a cosmetic singularity) of a holomorphic function is a point at which the function is undefined, but it is possible to define the function at that point in such a way that the function is regular in a neighbourhood of that point.
For instance, the function
has a singularity at z = 0. This singularity can be removed by defining f(0) := 1, which is the limit of f as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by f being given an indeterminate form. Taking a power series expansion for shows that
Formally, if is an open subset of the complex plane , a point of , and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on . We say is holomorphically extendable over if such a exists.
Riemann's theorem
Riemann's theorem on removable singularities states when a singularity is removable:
Theorem. Let be an open subset of the complex plane, a point of and a holomorphic function defined on the set . The following are equivalent:
- is holomorphically extendable over .
- is continuously extendable over .
- There exists a neighborhood of on which is bounded.
- .
The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at is equivalent to it being analytic at (proof), i.e. having a power series representation. Define
Clearly, h is holomorphic on D \ {a}, and there exists
by 4, hence h is holomorphic on D and has a Taylor series about a:
We have c0 = h(a) = 0 and c1 = hTemplate:'(a) = 0; therefore
Hence, where z≠a, we have:
However,
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:
- In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number such that . If so, is called a pole of and the smallest such is the order of . So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its poles.
- If an isolated singularity of is neither removable nor a pole, it is called an essential singularity. It can be shown that such an maps every punctured open neighborhood to the entire complex plane, with the possible exception of at most one point.