Removable singularity

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In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be so defined that it is continuous at the singularity.

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.

See also

sl:odpravljiva singularnost