principalpart

Principal part is a term used in linguistics and in complex analysis to describe a defining portion of a larger concept. In linguistics, it refers to a core set of verb forms from which other forms are derived. In mathematics, specifically in the study of complex functions, it denotes the portion of a Laurent series that encodes the function’s singular behavior at a point.

In historical languages such as Latin and Greek, the principal parts of a verb are a small set of its most important inflected forms used to derive the full conjugation. The exact set varies by language, but Latin commonly uses four parts to cover the main tenses and voices. Examples include amo, amare, amavi, amatum (to love); video, videre, vidi, visum (to see); fero, ferre, tuli, latus (to carry); esse, sum, esse, fui, futurus (to be). These principal parts allow a learner or grammarian to generate all other forms of the verb.

In complex analysis, the principal part of a Laurent series of a function f around an isolated singularity z0 is the finite sum of terms with negative powers of (z − z0). It reveals the nature of the singularity, such as the order of a pole. The coefficient of (z − z0)^(−1) in this part is the residue, a key quantity in contour integration. For example, if f(z) = 1/(z − 1)^3 + g(z) with g analytic at z = 1, the principal part about z0 = 1 consists of 1/(z − 1)^3, indicating a third-order pole.