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.

Read more about this topic:  Laurent Series

Famous quotes containing the words principal and/or part:

    Rather than have it the principal thing in my son’s mind, I would gladly have him think that the sun went round the earth, and that the stars were so many spangles set in the bright blue firmament.
    Thomas Arnold (1795–1842)

    He’d been numb a long time, years. All his nights down Ninsei, his nights with Linda, numb in bed and numb at the cold sweating center of every drug deal. But now he’d found this warm thing, this chip of murder. Meat, some part of him said. It’s the meat talking, ignore it.
    William Gibson (b. 1948)