Question about Elementary Functions

[lugita15]

Obviously, most elementary functions do not have elementary functions as integrals. For instance, the integral of e^-(x^2) is not an elementary function even though its integrand is. My question is, what is the smallest set of functions which includes the set of elementary function and is closed under integration?

 

[mathman]

To some extent it depends on a precise definition of "elementary function". For example, the integral of e^[(-x^2)/2] (with appropriate normalizations) is called erf(x). Whether or not it is elementary becomes a matter of definition.

 

[lugita15]

As to what constitutes an elementary function I define it as follows:
An elementary function is a function built from a finite number of exponentials, logarithms, trigonometric functions, inverse trigonometric functions, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ - × ÷). The roots of equations are the functions implicitly defined as solving a polynomial equation with constant coefficients.

 

[mathwonk]

even the simpler question of describing integrals of all elementary functions seems a tall order. i.e. of giving a collectioncontaining integrals of all elementarty functions and possibly not closed under integration.

i have no idea.

even the set of all elementary functions is a field extension of infinite transcendence degree over C(X) hence more than a bit complicated.