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Removable singularity

Undefined point on a holomorphic function which can be made regular

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

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

Riemann's theorem

Riemann's theorem on removable singularities is as follows:

f is holomorphically extendable over .
f is continuously extendable over .
There exists a neighborhood of a on which f is bounded.
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The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at a is equivalent to it being analytic at a (proof), i.e. having a power series representation. Define : h(z) = \begin{cases} (z - a)^2 f(z) & z \ne a ,\ 0 & z = a . \end{cases}

Clearly, is holomorphic on , and there exists : h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0 by 4, hence is holomorphic on and has a Taylor series about : : h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots , .

We have and ; therefore : h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots , .

Hence, where , we have: : f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \ldots , .

However, : g(z) = c_2 + c_3 (z - a) + \cdots , . is holomorphic on , thus an extension of .

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 m such that . If so, a is called a pole of f and the smallest such m is the order of . So removable singularities are precisely the poles of order . A meromorphic function blows up uniformly near its other poles.
If an isolated singularity a of f is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an f maps every punctured open neighborhood U \smallsetminus {a} to the entire complex plane, with the possible exception of at most one point.

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analytic-functions meromorphic-functions bernhard-riemann

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