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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
- <math> \text{sinc}(z) = \frac{\sin z}{z} </math>
has a singularity at Template:Math. This singularity can be removed by defining <math>\text{sinc}(0) := 1,</math> which is the limit of Template:Math as Template:Mvar tends to 0. The resulting function is holomorphic. In this case the problem was caused by Template:Math being given an indeterminate form. Taking a power series expansion for <math display="inline">\frac{\sin(z)}{z}</math> around the singular point shows that
- <math> \text{sinc}(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 theoremEdit
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 <math>a</math> is equivalent to it being analytic at <math>a</math> (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 <math> D \setminus \{a\}</math>, 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 c0 = h(a) = 0 and c1 = hTemplate:'(a) = 0; therefore
- <math>h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .</math>
Hence, where <math>z \ne a</math>, we have:
- <math>f(z) = \frac{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 <math> f </math>.
Other kinds of singularitiesEdit
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 of <math>f</math> and the smallest such <math>m</math> is the order of <math>a</math>. So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its other poles.
- If an isolated singularity <math>a</math> of <math>f</math> is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows 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.