Riemann's theorem on removable singularities is as follows: {{math theorem| Let D \subset \mathbb C be an open subset of the complex plane, a \in D a point of D and f a holomorphic function defined on the set {{tmath| D \smallsetminus \{a\} }}. The following are equivalent: • f is holomorphically extendable over . • f is continuously extendable over . • There exists a
neighborhood of a on which f is
bounded. • {{tmath|1= \lim_{z\to a}(z - a) f(z) = 0 }}.}} 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 : h(z) = \begin{cases} (z - a)^2 f(z) & z \ne a ,\\ 0 & z = a . \end{cases} Clearly, is holomorphic on {{tmath| D \smallsetminus \{a\} }}, and there exists : h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0 by 4, hence is holomorphic on and has a
Taylor series about : : h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, . We have and ; therefore : h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, . Hence, where , we have: : f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \ldots \, . However, : g(z) = c_2 + c_3 (z - a) + \cdots \, . is holomorphic on , thus an extension of . == Other kinds of singularities ==