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Saturday, February 04, 2006

Removable-singularity



In complex analysis, a removable singularity of a function (mathematics) is a point at which the function is not defined (a mathematical singularity) but at which the function can be defined without creating any problems. Problems would be discontinuity or non-differentiability. 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 LHopitals 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 U C which coincides with f on U - {a}. Such a holomorphic function g exists if and only if the limit (mathematics) limza f(z) exists, this limit is then equal to g(a). Riemanns theorem on removable singularities states that the singularity a is removable if and only if there exists a neighborhood (topology) of a on which f is bounded. The removable singularities are precisely the pole (complex analysis)s of order 0. See also: analytic capacity

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