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The principal part of a Laurent series

Definition

Let

$\sum_{n \in Z} c_{n} (z - a)^{n}$

be a convergent Laurent series on an annulus (Convergent Laurent series on an annulus). Its principal part is the negative-power subseries

$\sum_{m \geq 1} c_{- m} (z - a)^{- m} .$

Its regular part is the nonnegative-power subseries

$\sum_{n \geq 0} c_{n} (z - a)^{n} .$

Remarks

When the annulus is a punctured disc about $a$ , the principal part measures what fails to extend holomorphically across $a$ : it vanishes for a removable singularity, is finite and nonzero for a pole, and has infinitely many nonzero terms for an essential singularity. On an annulus with positive inner radius, $a$ is outside the domain and no singularity classification at $a$ is implied.

Depends on

Convergent Laurent series on an annulus

Used by

Dependency tree · two levels

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Sources

L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 5 §§1.1-1.3 (standard reference, not scraped)
Jeremy Orloff, MIT 18.04 Topic 7: Taylor and Laurent Series (standard reference, not scraped)