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{{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. 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}} | |||
Revision as of 22:14, 9 July 2013
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. 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 of a function
is the portion of the Laurent series consisting of terms with negative degree. That is,
is the principal part of at . has an essential singularity at , if and only if the principal part is an infinite sum.
Other definitions
Calculus
Consider the difference between the function differentialTemplate:Disambiguation needed and the actual increment:
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 distributions having a singular support at a single point.