Graph of x squared undefined at x equals 2.svg with a removable singularity at x = 2 ]] In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point. For instance, the (unnormalized) sinc function, as defined by has a singularity at z = 0 . This singularity can be removed by defining which is the limit of sinc as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by sinc being given an indeterminate form. Taking a power series expansion for around the singular point shows that Formally, if is an open subset of the complex plane \mathbb C, a \in U a point of U, and is a holomorphic function, then a is called a removable singularity for f if there exists a holomorphic function which coincides with f on. We say f is holomorphically extendable over U if such a g exists.

Riemann's theorem

Riemann's theorem on removable singularities is as follows: 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, and there exists by 4, hence h is holomorphic on D and has a Taylor series about a: We have c0 = h(a) = 0 and c1 = h'(a) = 0; therefore Hence, where z \ne a, we have: However, is holomorphic on D, thus an extension of f.

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: