Laurent Series - Principal Part

The principal part of a Laurent series is the series of terms with negative degree, that is

If the principal part of f is a finite sum, then f has a pole at c of order equal to (negative) the degree of the highest term; on the other hand, if f has an essential singularity at c, the principal part is an infinite sum (meaning it has infinitely many non-zero terms).

If the inner radius of convergence of the Laurent series for f is 0, then this is if and only if: f has an essential singularity at c if and only if the principal part is an infinite sum, and has a pole otherwise.

If the inner radius of convergence is positive, f may have infinitely many negative terms but still be regular at c, as in the example above, in which case it is represented by a different Laurent series in a disk about c.

Laurent series with only finitely many negative terms are tame—they are a power series divided by, and can be analyzed similarly—while Laurent series with infinitely many negative terms have complicated behavior on the inner circle of convergence.