Editing Removable singularity (section)

== Riemann's theorem ==

[[Bernhard Riemann|Riemann's]] theorem on removable singularities is as follows:

{{math theorem| Let <math>D \subset \mathbb C</math> be an open subset of the complex plane, <math>a \in D</math> a point of <math>D</math> and <math>f</math> a holomorphic function defined on the set <math>D \setminus \{a\}</math>.  The following are equivalent:

# <math>f</math> is holomorphically extendable over <math>a</math>. 
# <math>f</math> is continuously extendable over <math>a</math>. 
# There exists a [[neighborhood (topology)|neighborhood]] of <math>a</math> on which <math>f</math> is [[bounded function|bounded]].
# <math>\lim_{z\to a}(z - a) f(z) = 0</math>.}}

The implications 1 ⇒ 2  ⇒ 3  ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at <math>a</math> is equivalent to it being analytic at <math>a</math> ([[Proof that holomorphic functions are analytic|proof]]), i.e. having a power series representation. Define

:<math>
  h(z) = \begin{cases}
    (z - a)^2 f(z) &  z \ne a ,\\
    0              &  z = a .
  \end{cases}
</math>

Clearly, ''h'' is holomorphic on <math> D \setminus \{a\}</math>, and there exists
:<math>h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0</math>
by 4, hence ''h'' is holomorphic on ''D'' and has a [[Taylor series]] about ''a'':

:<math>h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .</math>

We have ''c''<sub>0</sub> = ''h''(''a'') = 0 and ''c''<sub>1</sub> = ''h{{'}}''(''a'') = 0; therefore

:<math>h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .</math>

Hence, where <math>z \ne a</math>, we have:

:<math>f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \cdots \, .</math>

However,

:<math>g(z) = c_2 + c_3 (z - a) + \cdots \, .</math>

is holomorphic on ''D'', thus an extension of <math> f </math>.