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In mathematics, the **principal part** has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function.

The **principal part** at of a function

is the portion of the Laurent series consisting of terms with negative degree.^{[1]} That is,

is the principal part of at . If the Laurent series has an inner radius of convergence of 0 , then has an essential singularity at , if and only if the principal part is an infinite sum. If the inner radius of convergence is not 0, then may be regular at despite the Laurent series having an infinite principal part.

Consider the difference between the function differential and the actual increment:

The differential *dy* is sometimes called the **principal (linear) part** of the function increment *Δy*.

The term **principal part** is also used for certain kinds of distributions having a singular support at a single point.

- Cauchy Principal Part at PlanetMath