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 [math]\displaystyle{ z=a }[/math] of a function

[math]\displaystyle{ f(z) = \sum_{k=-\infty}^\infty a_k (z-a)^k }[/math]

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

[math]\displaystyle{ \sum_{k=1}^\infty a_{-k} (z-a)^{-k} }[/math]

is the principal part of [math]\displaystyle{ f }[/math] at [math]\displaystyle{ a }[/math]. If the Laurent series has an inner radius of convergence of [math]\displaystyle{ 0 }[/math], then [math]\displaystyle{ f(z) }[/math] has an essential singularity at [math]\displaystyle{ a }[/math] if and only if the principal part is an infinite sum. If the inner radius of convergence is not [math]\displaystyle{ 0 }[/math], then [math]\displaystyle{ f(z) }[/math] may be regular at [math]\displaystyle{ a }[/math] despite the Laurent series having an infinite principal part.

Other definitions

Calculus

Consider the difference between the function differential and the actual increment:

[math]\displaystyle{ \frac{\Delta y}{\Delta x}=f'(x)+\varepsilon }[/math]

[math]\displaystyle{ \Delta y=f'(x)\Delta x +\varepsilon \Delta x = dy+\varepsilon \Delta x }[/math]

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