8. Theorem 2.3.1

Any holomorphic function $f : U \to C$ for $U = B_{R} (z_{0}) - \overset{―}{B_{r} (z_{0})}$ with $r < R$ , 1;; admits a unqiue Laurant expansion $\sum_{n = - \infty}^{\infty} c_{n} (z - z_{0})^{n}$ with $c_{n} = \oint \frac{f (z)}{(z - z_{0})^{n + 1}}$ .

Remark

To prove this, we start with a power series expansion around some point in the ring, and then morph it to get our desired result.