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

and lets , 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

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

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.

© 1996-9

1999-05-25