Principal part

This article is about the mathematical meaning. For the grammar term (a list of verb forms), see Principal parts.

In mathematics, the principal part has several independent meanings.

[edit] Laurent series

principal part of the Laurent series of a function f(z),

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

is the series of terms with negative degree, that is

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

If f(z) has an essential singularity at p, then the principal part is an infinite sum, and vice versa.

Consider the difference between the function differential 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 the principal (linear) part of the function increment Δy.

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

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