Definitions

Principal part

Principal part

In mathematics, the principal part has several independent meanings.

Laurent series

principal part of the Laurent series of a function f(z),
sum_{k=-infty}^infty a_k (z-a)^k
is the series of terms with negative degree, that is
sum_{k=-infty}^{-1} a_k (z-a)^k

If f(z) has an essential singularity at p, then the principal part is an infinite sum, and vice versa.

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 the principal (linear) part of the function increment Δy.

Distribution theory

The term principal part is used for certain kinds of distribution having a singular support at a single point.

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