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 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 ${\displaystyle 0}$, 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 ${\displaystyle 0}$, 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.

See also

• Mittag-Leffler's theorem
• Cauchy principal value