Elementary Function

A function built up of compositions of algebraic functions, the Exponential Function and the Trigonometric Functions and their inverses by Addition, Multiplication, Division, root extractions (the Elementary Operations) under repeated compositions (Shanks 1993, p. 145). Unfortunately, there are several different definitions of what constitutes an elementary function.

Following Liouville, Watson (1966, p. 111) defines

$\displaystyle l_1(z)$	$\textstyle \equiv$	$\displaystyle l(z)\equiv \ln(z)$
$\displaystyle e_1(z)$	$\textstyle \equiv$	$\displaystyle e(z)\equiv e^z$
$\displaystyle \varsigma_1 f(z)$	$\textstyle \equiv$	$\displaystyle \varsigma f(z) \equiv \int f(z)\,dz,$

and lets $l_2\equiv l(l(z))$ , etc. These functions are then called elementary, although Watson confusingly terms them ``elementary transcendental functions.''

Not all functions are elementary. For example, the Normal Distribution Function

$\begin{displaymath} \Phi(x) \equiv {1\over\sqrt{2\pi}} \int^x_0 e^{-t^2/2}\, dt \end{displaymath}$

is a notorious example of a nonelementary function. The Elliptic Integral

$\begin{displaymath} \int \sqrt{1-x^4}\,dx \end{displaymath}$

is another. Nonelementary functions are called Transcendental Functions.

References

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, p. 111, 1966.