Existence of Laurent Series

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

Let $R \in \R_{>0}$ be a real number.

Let $\map {B'} {z_0, R}$ be the open punctured disk at $z_0$ of radius $R$.

Let $f: \map {B'} {z_0, R} \to \C$ be holomorphic.

Then there exists a sequence $\sequence {a_n}_{n \mathop \in \Z}$ such that:
 * $\map f z = \displaystyle \sum_{n = -\infty}^\infty a_n \paren {z - z_0}^n$

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

Proof
Choose circles $C_1$ and $C_3$ centered at $z_0$ and connect them by path $C_2$ such that $z$ is inside $C_1 + C_2 - C_3 - C_2$ as shown below:


 * Path figure.png

This curve and its interior are contained in $B'$, so by Cauchy's Integral Formula:

Since $C_1$ contains $z$ we have for all $w$ on $C_1$:


 * $\dfrac {\cmod {z - z_0} } {\cmod {w - z_0} } < 1$

Therefore:

Similarly, since $z$ lies outside $C_3$ we have for all $w$ on $C_3$:


 * $\dfrac {\cmod {w - z_0} } {\cmod {z - z_0} } < 1$

Therefore:

Combining what has been shown above yields: