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Principal part

## Summary

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 ${\displaystyle z=a}$  of a function

${\displaystyle f(z)=\sum _{k=-\infty }^{\infty }a_{k}(z-a)^{k}}$

is the portion of the Laurent series consisting of terms with negative degree.[1] That is,

${\displaystyle \sum _{k=1}^{\infty }a_{-k}(z-a)^{-k}}$

is the principal part of ${\displaystyle f}$  at ${\displaystyle a}$ . If the Laurent series has an inner radius of convergence of 0 , then ${\displaystyle f(z)}$  has an essential singularity at ${\displaystyle a}$ , if and only if the principal part is an infinite sum. If the inner radius of convergence is not 0, then ${\displaystyle f(z)}$  may be regular at ${\displaystyle a}$  despite the Laurent series having an infinite principal part.

## Other definitions

### Calculus

Consider the difference between the function differential and the actual increment:

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

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

### Distribution theory

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