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# Principal part

In mathematics, the principal part has several independent meanings.

## Laurent series

principal part of the Laurent series of a function f(z),
$sum_\left\{k=-infty\right\}^infty a_k \left(z-a\right)^k$
is the series of terms with negative degree, that is
$sum_\left\{k=-infty\right\}^\left\{-1\right\} a_k \left(z-a\right)^k$

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

## Calculus

Consider the difference between the function differential and the actual increment:
$frac\left\{Delta y\right\}\left\{Delta x\right\}=f\text{'}\left(x\right)+varepsilon$
$Delta y=f\text{'}\left(x\right)Delta x +varepsilon Delta x = dy+varepsilon Delta x$
The differential dy is the principal (linear) part of the function increment Δy.

## Distribution theory

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