 Principal Part
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Principal Part

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 ?y.

### Distribution theory

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