Nonanalytically

Nonanalytically is an adverb describing an action taken without the use of analytic methods or expressions, or a property that fails to be analytic. It derives from the adjective nonanalytic with the adverbial suffix -ly. In mathematics, a function is analytic at a point if it equals its Taylor series in a neighborhood; many functions are not analytic there, and nonanalytic behavior is described as nonanalytic. The term nonanalytically is used when contrasting an approach that relies on analytic tools (such as closed-form formulas, power series, or complex-analytic techniques) with one that does not. Examples include solving differential equations numerically rather than obtaining a closed-form solution, or analyzing a function by qualitative or numerical means rather than invoking an analytic representation.

In physics and applied sciences, nonanalytic behavior refers to functions that are not analytic across a parameter domain—such as phase transitions where thermodynamic potentials become nonanalytic at critical points, or certain response functions with sharp jumps. The term may be found in academic prose to emphasize methods or results that rely on computation, simulation, or empirical observation rather than symbolic manipulation.

While nonanalytic methods are common in numerical analysis and experimental work, the formal notion of nonanalyticity is defined rigorously within analysis, while nonanalytically is a stylistic expression of approach.