Principal part

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 {\displaystyle z=a} of a function

f ( z ) = ∑ k = − ∞ ∞ a k ( z − a ) k {\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,

∑ k = 1 ∞ a − k ( z − a ) − k {\displaystyle \sum _{k=1}^{\infty }a_{-k}(z-a)^{-k}}

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

Δ y Δ x = f ′ ( x ) + ε {\displaystyle {\frac {\Delta y}{\Delta x}}=f'(x)+\varepsilon }
Δ y = f ′ ( x ) Δ x + ε Δ x = d y + ε Δ x {\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

References

  1. ^ Laurent. 16 October 2016. ISBN 9781467210782. Retrieved 31 March 2016.

External links

  • Cauchy Principal Part at PlanetMath