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.

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.

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.

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

• Mittag-Leffler's theorem
• Cauchy principal value

• Cauchy Principal Part at PlanetMath.