Principal part

Short description: Widely-used term in mathematics

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

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

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

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

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:

$\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=f^{\prime }(x)+\epsilon$

$\mathrm{\Delta }y=f^{\prime }(x)\mathrm{\Delta }x+\epsilon \mathrm{\Delta }x=dy+\epsilon \mathrm{\Delta }x$

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

External links

• Cauchy Principal Part at PlanetMath