7. Definition 2.3.5 - Laurant Series

A Laurant series is a double power series:

\sum_{n = - \infty}^{\infty} c_{n} (z - z_{0})^{n} = \sum_{n = 0}^{\infty} c_{n} (z - z_{0})^{n} + \sum_{n = - \infty}^{- 1} c_{n} (z - z_{0})^{n}

The postive part of the series is called the regular part, and the negative part is the principal part. We say that a Laurant Series converges if 1;; the regular part and the principal part converge.

Recall that we already know that the regular part of the Laurant Series has some convergence radius, say $R$ .

In addition, we can view the principal part as

\sum_{n = 1}^{\infty} c_{- n} {(\frac{1}{z - z_{0}})}^{n}

Which also converges for some $| \frac{1}{z - z_{0}} | < r$ and thus $\frac{1}{r} < | z - z_{0} |$ . Thus we get that the Laurant series is convergent for $\frac{1}{r} < | z - z_{0} | < R$ . In general, the domain of convergence of a Laurant series forms a ring.