Elementary Function with Non-elementary Derivative

[pierce15]

Does such a function exist? My gut tells me that such a function should not exist, but is there a proof that all elementary functions have elementary derivatives?

 

[mathman]

Does such a function exist? My gut tells me that such a function should not exist, but is there a proof that all elementary functions have elementary derivatives?

I believe that you could make a finite list of all elementary functions and then show that they all have derivatives.

 

[AlephZero]

I think this is a question about language not math, because "elementary functions" are whatever functions people decide to call "elementary". AFAIK the phrase "elementary function" doesn't have any mathematical siignificance (unlike "analytic function", for example).

A common reason for inventing a new "elementary" function is to give a name to the indefinite integral of some function that has a "practical" use (in physics or engineering) - which explaiins why most (if not all) of them have elementary derivatives.

 

[pwsnafu]

I think this is a question about language not math, because "elementary functions" are whatever functions people decide to call "elementary".

Elementary functions were introduced by Liouville. They are well defined.

AFAIK the phrase "elementary function" doesn't have any mathematical significance (unlike "analytic function", for example).

It is currently the largest function space on which the Risch algorithm is known to be decidable.

 

[JJacquelin]

A common reason for inventing a new "elementary" function is to give a name to the indefinite integral of some function that has a "practical" use (in physics or engineering)

Roughly I agree, but with some nuances.
First, I prefer to say : A common reason for inventing a new "special" function is to give a name to the indefinite integral ... etc.
Second, if a "special" function becomes of common use and is known by a very large number of people, it can be said "elementary" function in the common language.
Third, giving a name to the integral of a already known function is not the only way to invent a new special function. Many were defined as a solution of a differential equation, or as a solution of an analytic equation (for example the W Lambert function), or thanks to several other kind of definitions. For example, see pp. 20-25 in the paper "Safari in the Contry of Special Functions" :
http://www.scribd.com/JJacquelin/documents

 

[pierce15]

http://www.scribd.com/JJacquelin/documents

Impressive set of documents! I have a question about the sophomore dream: how would one show that:

$$ \int_0^1 x^{-x} \, dx = \sum_{n=1}^\infty \frac{1}{n^n} $$

 

[JJacquelin]

Hi piercebeatz !

See attachment.
Also, see : http://mathworld.wolfram.com/SophomoresDream.html