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Removable singularity

In complex analysis, a removable singularity of a function is a point at which the function is not defined (i.e. a singularity) but at which the function can be defined without creating any problems.

For instance, the function f(z) = sin(z)/z for z ≠ 0 has a removable singularity at z = 0: we can define f(0) = 1 and the resulting function will be continuous and even differentiable (a consequence of L'Hopital's rule).

Formally, if U is an open subset of the complex plane C, a is an element of U and f : U - {a} → C is a holomorphic function, then z is called a removable singularity for f if there exists a holomorphic function g : UC which coincides with f on U - {a}. Such a holomorphic function g exists if and only if the limit limza f(z) exists; this limit is then equal to g(a).

Riemann's theorem on removable singularities states that the singularity a is removable if and only if there exists a neighborhood of a on which f is bounded.

The removable singularities are precisely the poles of order 0.



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