Removable singularity
|
|
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 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 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 : U → C which coincides with f on U - {a}. Such a holomorphic function g exists if and only if the limit limz→a 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.