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Removable singularity
Undefined point on a holomorphic function which can be made regular / From Wikipedia, the free encyclopedia
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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.
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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.