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Concept
Removable singularity
Résumé
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
has a singularity at z = 0. This singularity can be removed by defining 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 around the singular point 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 on removable singularities is as follows:
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 , 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 = h(a) = 0; therefore
Hence, where , we have:
However,
is holomorphic on D, thus an extension of .
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 .
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