Principal part

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 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.

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.