Elementary Functions - What Is The Exact Definition?

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• diegogarcia
In summary: Pre-calculus".So, at different schools, the name of the course might have been different, but the content was the same.In summary, the elementary functions are a set of functions that are not differentiated. They include polynomial, logarithmic, exponential, trigonometric, and hyperbolic functions, and their inverses.
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.

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
diegogarcia said:
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.

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
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!

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.

Liouville's theorem is not elementary!

The top of the elementary functions page has the definition, the part about differential algebras is just trying to extend the normal definition to other contexts.

The name of a course at many schools at one time was called "Pre-Calculus" and was alternatively called "College Algebra And Trigonometry". Some years later, this same course was called, at some schools, "Elementary Functions"

1. What are elementary functions?

Elementary functions are basic mathematical functions that can be expressed using a finite number of algebraic operations, such as addition, subtraction, multiplication, division, and raising to a power. These functions include polynomials, exponential functions, logarithmic functions, and trigonometric functions.

2. What is the exact definition of an elementary function?

The exact definition of an elementary function is a function that can be defined using a finite number of elementary operations, such as addition, subtraction, multiplication, division, and exponentiation, along with algebraic constants and variables. This means that the function can be written using a combination of these operations and does not involve any complicated or non-elementary functions.

3. How are elementary functions different from non-elementary functions?

Elementary functions are limited to a finite number of basic operations and can be easily expressed using algebraic equations. Non-elementary functions, on the other hand, involve more complex operations, such as integration, differentiation, and special functions like the gamma function or Bessel function. These functions cannot be expressed using a finite number of elementary operations.

4. What are some examples of elementary functions?

Some examples of elementary functions include linear functions, quadratic functions, exponential functions, logarithmic functions, trigonometric functions (sine, cosine, tangent), and their inverse functions. Other examples include polynomial functions, rational functions, and power functions.

5. Why are elementary functions important in mathematics?

Elementary functions are important in mathematics because they form the building blocks for more complex functions. They are used extensively in calculus, differential equations, and other areas of mathematics to model and solve real-world problems. Understanding elementary functions is crucial for students to progress to higher levels of math and to apply mathematical concepts in various fields, such as physics, engineering, and economics.

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