nonanalytic

Nonanalytic is an adjective used in several scholarly fields to indicate the absence of an analytic property as defined within a given framework.

In mathematics, a function is analytic at a point if it can be expressed by a convergent power series in a neighborhood of that point. A nonanalytic function does not admit such a representation. This distinction is stronger than mere differentiability: a function can be infinitely differentiable (smooth) but still fail to be analytic. A classic real example is the function defined by f(x) = e^{-1/x^2} for x ≠ 0 and f(0) = 0, which is smooth at 0 but not analytic there. In complex analysis, analytic functions are precisely the holomorphic functions on a domain; a nonanalytic function is not holomorphic anywhere in its domain (for instance, the function z ↦ z̄ is not analytic anywhere in the complex plane).

In linguistics and related fields, nonanalytic is used less uniformly. It may appear in contrast to analytic languages, or more broadly to describe forms or constructions that rely more on morphology and inflection rather than fixed word order and auxiliary words. Because analytic versus nonanalytic usage varies by subfield and authors, the label is not as standardized as analytic versus synthetic or fusional versus agglutinative classifications.

See also: analytic, holomorphic, Taylor series, smooth function, linguistic typology.