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{{Short description|Widely-used term in mathematics}} {{about|the mathematical meaning|the grammar term (a list of verb forms)|Principal parts}} 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 definition== The '''principal part''' at <math>z=a</math> of a function : <math>f(z) = \sum_{k=-\infty}^\infty a_k (z-a)^k</math> is the portion of the [[Laurent series]] consisting of terms with negative degree.<ref>{{cite book | url=https://books.google.com/books?id=_cADk52kr4oC&dq=%22is+the+portion+of+the+Laurent+series+consisting+of+terms+with+negative+degree.%22&pg=PT48 | title=Laurent | date=16 October 2016 | isbn=9781467210782 | accessdate=31 March 2016}}</ref> That is, : <math>\sum_{k=1}^\infty a_{-k} (z-a)^{-k}</math> is the principal part of <math>f</math> at <math> a </math>. If the Laurent series has an inner radius of convergence of <math>0</math>, then <math>f(z)</math> has an [[essential singularity]] at <math>a</math> if and only if the principal part is an infinite sum. If the inner radius of convergence is not <math>0</math>, then <math>f(z)</math> may be regular at <math>a</math> despite the Laurent series having an infinite principal part. ==Other definitions== ===Calculus=== Consider the difference between the function [[differential of a function|differential]] and the actual increment: :<math>\frac{\Delta y}{\Delta x}=f'(x)+\varepsilon </math> :<math> \Delta y=f'(x)\Delta x +\varepsilon \Delta x = dy+\varepsilon \Delta x</math> The differential ''dy'' is sometimes called the '''principal (linear) part''' of the function increment ''Ξy''. ===Distribution theory=== The term '''principal part''' is also used for certain kinds of [[distribution (mathematics)|distributions]] having a [[singular support]] at a single point. ==See also== *[[Mittag-Leffler's theorem]] *[[Cauchy principal value]] ==References== {{Reflist}} ==External links== *[http://planetmath.org/encyclopedia/CauchyPrinciplePartIntegral.html Cauchy Principal Part at PlanetMath] [[Category:Complex analysis]] [[Category:Generalized functions]]
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