📂Complex Anaylsis

The Principal Part of Laurent Series and Classification of Singularities

Overview ¹

If we closely examine the principal part of the Laurent expansion, we can identify the type of singularities.

Let’s say $\alpha$ is an isolated singularity of the 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]: For all $n$, if $b_{n}=0$ $ \iff$ $\alpha$ is a removable singularity.
• [2]: For some $m$, if $b_{m} \ne 0$ and $b_{m+1} = b_{m+2} = \cdots = 0$ $\iff$ $\alpha$ is a $m$-order pole.
• [3]: Not all $k$, but there exists an infinite number of $k$ that satisfies $b_{k} \ne 0$. $\iff$ $\alpha$ is an essential singularity.

Explanation

The proof is not very important, and the facts are not bad to know. However, sometimes these facts can be helpful, so if you have time, memorize them; if not, just be aware that such facts exist.