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## 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 $z=a$  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,

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

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

## Other definitions

### Calculus

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 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.