Principal part


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

Laurent series definitionEdit

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.

Other definitionsEdit


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.

Distribution theoryEdit

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

See alsoEdit


  1. ^ Laurent. 16 October 2016. ISBN 9781467210782. Retrieved 31 March 2016.

External linksEdit

  • Cauchy Principal Part at PlanetMath