📂Complex Anaylsis

The Principal Part of the Laurent Series and Classification of Singularities

Overview¹

By carefully examining the principal part of a Laurent expansion, one can determine the type of a singularity.

Let $\alpha$ be an isolated singularity of a function $f:A\subset \mathbb{C} \to \mathbb{C}$. For its Laurent expansion $$ f(z) = \sum_{n = 0 }^{\infty} a_{n} (z-\alpha) ^{n} + \sum_{n = 1 }^{\infty} { {b_{n} } \over{ (z-\alpha) ^{n} } } $$ the sequence $b_{n}$ has the following properties.

Theorem

• [1]: $b_{n}=0$ for all $n$ $ \iff$ $\alpha$ is a removable singularity.
• [2]: $b_{m} \ne 0$ for some $m$ and $b_{m+1} = b_{m+2} = \cdots = 0$ $\iff$ $\alpha$ is a pole of order $m$.
• [3]: There exist infinitely many $k$ satisfying $b_{k} \ne 0$, though not for all $k$. $\iff$ $\alpha$ is an essential singularity.

Explanation

The proof is not particularly important, and even the facts themselves are only worth knowing to a modest degree. Still, since these facts occasionally turn out to be helpful, memorize them if you have the time, and if not, just keep in mind that such facts exist.