# Elementary Functions - What Is The Exact Definition?

diegogarcia
TL;DR Summary
Is there an exact definition for the term "elementary function?"
Mathematicians will use the term "elementary functions," often in the context of integration wherein some integrals cannot be expressed in elementary functions.

The elementary functions are usually listed as being arithmetic, rational, polynomial, exponential, logarithmic, trigonometric, hyperbolic, and their inverses.

But why just these? Why are not the special functions, like elliptic integrals, the hypergeometric functions, etc., also included? After all, the logarithmic and trigonomteric functions are defined (or certainly can be defined) as integrals just like the special functions. So why are the special functions like elliptic integrals excluded from the list of elementary functions.

After a bit of searching I find this possible definition for the elementary functions:

https://en.wikipedia.org/wiki/Elementary_function#Differential_algebra

I am not versed in abstract algebra but can this be the exact definition for an elemantary function?

If so, and even though I don't understand how, it must be the reason why the special functions and other functions are excluded.

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Your list of elementary functions have so many uses that they are well known by high school math students. That can not be said for the special functions that you listed.

symbolipoint and fresh_42
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If so, and even though I don't understand how, it must be the reason why the special functions and other functions are excluded.
Because they are not elementary. Special functions, for example, cannot easily be differentiated.

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Hi, @diegogarcia
Special Functions Arising from Integrals
The integrals $$\displaystyle\int\dfrac{dx}{x}=\ln x+C\quad\mbox{and}\quad\displaystyle\int\dfrac{dx}{1+x^2}=\tan^{-1}x+C$$
both take algebraic functions to a function that is not produced by adding, substracting, multiplying, or dividing.
The functions (...) mostly come from a class called Elementary Functions, which consist of polynomials, logarithms, exponentials, trigonometric and hyperbolic functions, and their inverses and also finite sums, differences, products, quotients, powers, and roots of such functions. The derivative of any differentiable elementary function is elementary, but an integral may or may not be elementary. This expands the class of functions to (...) Special Functions, like Bessel function, the Error Function.
Hope helps.
Regards

symbolipoint and PeroK
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Elementary function isn't a well-defined name. The least property should be that they are not defined by integrals themselves. This rules out Bessel-, Gamma-, Li-, and Si-functions. My first thought was: everything that is on a standard calculator, but this would translate to everything with a Taylor series. Maybe it would be better to say: everything that can be explained by compass and ruler!

diegogarcia
I believe that the answer is here:

https://en.wikipedia.org/wiki/Liouville's_theorem_(differential_algebra)#Examples

If I understand this correctly the integral of 1/(x^2+1) dx, which defines the arctan(x), would not qualify as being an "elementary" function, and hence neither would the inverse, which produces the trig functions.

However, Liouville's theorem, in conjunction with Euler's formula, can express this result (arctan(x)) as a logarithm of rational functions and it is this fact that make arctan(x) elementary (and also its inverse and hence all the trig functions).

This then gives an explicit and rigorous definition of an "elementary" function in terms of a differential algebra or differential field.

As I mentioned, I am not versed in abstract algebra but it seems that the proof exists that the "elementary" functions are indeed limited to those previously listed.

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Liouville's theorem is not elementary!

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