Nonelementary

Nonelementary is a mathematical term used to describe objects that do not belong to an elementary class. The precise meaning depends on the context, but the common usage indicates a higher level of complexity or structure than what is captured by the elementary category.

In the context of Möbius or Kleinian groups, a subgroup of PSL(2, C) is elementary if its action on the Riemann sphere has at most two limit points. A nonelementary subgroup has a limit set with at least three points, is infinite, and is not virtually abelian. Such groups exhibit richer dynamics and often contain free subgroups; examples include Schottky groups, whose limit sets are fractal in nature.

In calculus and differential algebra, a function or antiderivative is elementary if it can be expressed by a finite combination of algebraic operations, exponentials, logarithms, and trigonometric functions. A nonelementary integral cannot be expressed in such a form. Classical examples include the integral of e^{-x^2}, whose antiderivative is the error function, and many other integrals encountered in Liouville’s theory. The question of nonelementarity is studied through results like Liouville’s theorem and computational tools such as the Risch algorithm.

The term nonelementary is also used more broadly to denote non-elementary objects in other mathematical settings, signaling greater complexity than the elementary class. See also: elementary function, elementary group, Kleinian group, Schottky group, Liouville’s theorem.