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.^[1] That is,

$\sum_{k=-\infty}^{-1} a_k (z-a)^k$

is the principal part of $f$ at $a$. $f(z)$ has an essential singularity at $a$, if and only if the principal part is an infinite sum.

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.

See also

• Mittag-Leffler's theorem
• Cauchy principal value

References

1. "laurent". Retrieved 31 March 2016.

External links

• Cauchy Principal Part at PlanetMath