In mathematics, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and nth roots through composition and combinations using the four elementary operations. By allowing these functions to be complex numbers, trigonometric functions and their inverses are included in the elementary functions.The roots of equations are the functions implicitly defined as solving a polynomial equation with constant coefficients. For polynomials of degree four and smaller there are explicit formulas for the roots, but even for higher degree polynomials the fundamental theorem of algebra and the implicit function theorem assures the existence of a function that returns each one of the roots of a polynomial equation.Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s.

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