Principal part

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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[edit]

The principal part at z=a of a function

f(z) = \sum_{k=-\infty}^\infty a_k (z-a)^k

is the portion of the Laurent series consisting of terms with negative degree. That is,

\sum_{k=-\infty}^{-1} a_k (z-a)^k

is the principal part of f at  a . f(z) has an essential singularity at a, if and only if the principal part is an infinite sum.

Other definitions[edit]

Calculus[edit]

Consider the difference between the function differential[disambiguation needed] and the actual increment:

\frac{\Delta y}{\Delta x}=f'(x)+\varepsilon
 \Delta y=f'(x)\Delta x +\varepsilon \Delta x = dy+\varepsilon \Delta x

The differential dy is sometimes called the principal (linear) part of the function increment Δy.

Distribution theory[edit]

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

See also[edit]

External links[edit]