In mathematics, the principal part has several independent meanings but usually refers to the negative-power portion of the Laurent series of a function.

The principal part at ${\displaystyle z=a}$ of a function

is the portion of the Laurent series consisting of terms with negative degree. That is,

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

Consider the difference between the function differential and the actual increment:

The differential dy is sometimes called the principal (linear) part of the function increment Δy.

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