Principal part

Definition

Let $c \in \mathbb{C}$, let $R \in (0, \infty]$, and let $f$ be holomorphic on the punctured disc $0 < |z - c| < R$. Let $f(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n$ be the Laurent expansion of $f$ on this punctured disc. The principal part of $f$ at $c$ is the series $$\sum_{n = 1}^{\infty} a_{-n} (z - c)^{-n},$$ the part of the Laurent series with the negative powers of $z - c$.

Remarks

The principal part converges for each $z \neq c$. The difference $f(z) - \sum_{n \ge 1} a_{-n} (z - c)^{-n}$ extends to a holomorphic function on $|z - c| < R$. The principal part is $0$ at a removable singularity, it has finitely many nonzero terms at a pole, and it has infinitely many at an essential singularity. For example, the principal part of $\frac{e^z}{z^2}$ at $0$ is $\frac{1}{z^2} + \frac{1}{z}$.

Explanations

L1 human

The principal part is the piece of the function that is responsible for its bad behavior at $c$. If you subtract it, what is left is holomorphic at $c$.