A function composed of constants and basic elementary functions through four operations of finite degree and composite steps of finite degree functions can be expressed by a formula.

In mathematics, an elementary function is a univariate (usually a real number or a complex number) function, which is defined as the sum, product, root and combined function of a finite number of polynomials, rational numbers, trigonometric functions, hyperbola and exponential functions (possibly including their inverse functions) (such as arcsine, log or x 1/n).

All elementary functions are continuous in their domain.

Basic example

All functions obtained by adding, subtracting, multiplying or dividing any previous function in a finite number.

All functions obtained by finding the roots of the polynomial of elementary function coefficients.

All functions composed of a limited number of previously listed functions.

Some basic functions of a single complex variable z, in addition, others may get some function categories by using the latter two rules. For example, the exponential function e z {\ displaystyle e {z}} composed of addition, subtraction and division provides a hyperbolic function, while the initial combination zi {\ displaystyle z {i}} provides a trigonometric function.

An example of a non-elementary function is an error function.

A fact that may not be obvious, but it can be proved by Risch algorithm.