Principal part

Definition

Suppose $U\subset \mathbb{C}$ is an open subset and $z_0\in U$. Suppose $f:U\setminus z_0\to \mathbb{C}$ is a holomorphic function, so $z_0$ is an isolated singularity of $f$. The principal part of $f$ at $z_0$ is defined in the following equivalent ways:

• It is the function given in a neighborhood of $z_0$ by the part of the Laurent series for $f$ about $z_0$, comprising only the negative powers.
• It is the unique function $g$ (upto germ equivalence at $z_0$) such that $f-g$ is a holomorphic function, and such that $g$ has no nonnegative powers in its Laurent series expansion.