Existence of Laurent Series

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Theorem

Let $z_0 \in \C$ be a complex number.

Let $R>0$ be a real number.

Let $B'(z_0, R)$ be the open punctured disk at $z_0$ of radius $R$.

Let $f : B'(z_0, R) \to \C$ be holomorphic.


Then there exists a sequence $(a_n)_{n\in\Z}$ such that:

$f(z) = \displaystyle \sum_{n = -\infty}^\infty a_n \left({z - z_0}\right)^n$

for all $z \in B'(z_0, R)$.


Proof