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 U is an open subset of the complex plane C, a is a point of U, and f: U − {a} → C is a holomorphic function, then a is called a removable singularity for f if there exists a holomorphic function g: U → C which coincides with f on U − {a}. We say f is holomorphically extendable over U if such a g exists.
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Riemann's theorem
Riemann's theorem on removable singularities states when a singularity is removable:
Theorem. Let D be an open subset of the complex plane, a a point of D and f a holomorphic function defined on the set D \ {a}. The following are equivalent:
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
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 a_{0} = h(a) = 0 and a_{1} = h'(a) = 0, therefore
is a holomorphic extension of f over a, which proves the claim.
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:
See also
External links
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