has a singularity at z = 0. This singularity can be removed by defining , which is the limit of as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by 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, a point of , and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on . We say is holomorphically extendable over if such a exists.
The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at is equivalent to it being analytic at (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
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:
In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number such that . If so, is called a pole of and the smallest such is the order of . So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its other poles.
If an isolated singularity of is neither removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an maps every punctured open neighborhood to the entire complex plane, with the possible exception of at most one point.