Free-radical halogenation: Difference between revisions

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{{about|the mathematical meaning|the grammar term (a list of verb forms)|Principal parts}}
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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.
==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. That is,
: <math>\sum_{k=-\infty}^{-1} a_k (z-a)^k</math>
is the principal part of <math>f</math> at <math> a </math>.
<math>f(z)</math> has an [[essential singularity]] at <math>a</math>, if and only if the principal part is an infinite sum.
 
==Other definitions==
===Calculus===
Consider the difference between the function [[differential (mathematics)|differential]]{{disambiguation needed|date=July 2013}} 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 ''Δy''.
===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]]
 
==External links==
*[http://planetmath.org/encyclopedia/CauchyPrinciplePartIntegral.html Cauchy Principal Part at PlanetMath]
 
[[Category:Complex analysis]]
[[Category:Generalized functions]]
 
 
{{mathanalysis-stub}}

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