The principal part at an isolated singularity

Let $a\in \mathbb{C}$, let $R>0$, and let $f$ be holomorphic on the punctured disc $0<\mathrm{\mid }z-a\mathrm{\mid }<R$. By Laurent expansion on an annulus, the function $f$ has a Laurent expansion about $a$ on that punctured disc. By Laurent series split into regular and principal parts, this expansion splits uniquely into its regular and principal parts.

The principal part of $f$ at $a$ is the negative-power part of that Laurent expansion:

${\sum }_{m\ge 1}{c}_{-m}(z-a{)}^{-m}\mathrm{.}$

For Mittag-Leffler data, a prescribed principal part at $a$ means a finite sum

$p_a(z)={\sum }_{m=1}^M{c}_{-m}(z-a{)}^{-m}\mathrm{.}$

This is exactly the shape of the principal part of a pole whose order is at most $M$, and its order is exactly $M$ when $c_{-M}\mathrm{\ne }0$.

The point of the adjective "prescribed" is that one starts with the negative Laurent polynomial and asks for a meromorphic function having it at $a$. The pole order is then the largest $m$ with $c_{-m}\mathrm{\ne }0$.