- #1

diegogarcia

- 16

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- 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.

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.