Principal part

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. That is, : \sum_{k=1}^\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{\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.

References

References

1. (16 October 2016). "Laurent".