Principal part
Template:Short description Template:About 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 <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>Template:Cite book</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 definitionsEdit
CalculusEdit
Consider the difference between the 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 Δy.
Distribution theoryEdit
The term principal part is also used for certain kinds of distributions having a singular support at a single point.