Principal part explained

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 functionf(z) = \sum ^{inf ty} _{k =-inf ty} a _{k} (z - a) ^{k} is the portion of the Laurent series consisting of terms with negative degree.^[1] That is,\sum ^{inf ty} _{k =1} 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:

(\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

External links

• Cauchy Principal Part at PlanetMath

Notes and References

1. Book: Laurent . 16 October 2016 . 9781467210782 . 31 March 2016.