1. ## integral e^x2

It has been parroted by many that $\int e^{x^2}~dx$ does not have a solution in terms of any combination of the elementary functions.

Prove it! prove no such combination exists!

or prove the series expansion which is the solution cannot be expressed as such a combination.

2. Originally Posted by kwestor
It has been parrotted by many that the integral of e^x2 does not have a solution in terms of any combination of the elementary functions.

Prove it! prove no such combination exists!

or prove the series expansion which is the solution cannot be expressed as such a combimation.

yawwwwn ... another one.

Integration of Nonelementary Functions

3. Double Yawwnnnn!
Nope! all that link does is say one "should" be able to prove it using Liouville's Theorem.

it Doesn't actully show any such proof.

can you show me How the theorem is to be used to prove int(e^x2) not elementary?

nor does the link even derive Liouvilles Theorem in the first place so that would ALSO be necessary.

4. Stop yawning, guys... You make me want to go to bed

This link provides an explanation, but advanced, since it uses advanced algebra : http://www.fimfa.ens.fr/fimfa/IMG/Fi...tharoubane.pdf (it's of MSc level)

And I'm sorry, it's in French. I think that parts 3.2 and 3.3 are exactly what you want to disprove.
(Liouville-Ostrowski theorem)

I'm not completely sure whether the link is good, because it's a field in mathematics that I don't know at all. But the school from which these students come is maybe the best in France, regarding mathematics.

I can translate it but you would have to give me some time, or someone else can do it (clic-clac looks good at algebra and would certainly be better at some subtleties).

I found this too : Liouville's theorem on functions with elementary integrals.

5. $f(x)=e^x$ is not a elementary function. But we still easily express and use, even integrate. It means we need construct another theory in order to integrate. It looks like to find calculus to solve 'impossible problem' of 400 years ago.
I think human limitation is NOT 'impossible'!

I do something http://www.mathhelpforum.com/math-he...-function.html

6. Originally Posted by kwestor
Double Yawwnnnn!
Nope! all that link does is say one "should" be able to prove it using Liouville's Theorem.

it Doesn't actully show any such proof.

can you show me How the theorem is to be used to prove int(e^x2) not elementary?

nor does the link even derive Liouvilles Theorem in the first place so that would ALSO be necessary.
This reminds me of the guy who wanted to cut a big lump of wood in half. I give the guy a saw. The guy says: "But the wood isn't cut in half".

I feel like triple yawning when I say that when someone hands you the saw, it's your job to start sawing .... (and ask for help only when you're stuck).

On top of all this, you not only want someone to do the sawing for you, you want that person to actually make the saw as well .... I wonder how much of the saw making process you would actually understand ....? (and therefore how worhtwhile the whole exercise actually is).

7. Originally Posted by math2009
$f(x)=e^x$ is not a elementary function. But we still easily express and use, even integrate. It means we need construct another theory in order to integrate. It looks like to find calculus to solve 'impossible problem' of 400 years ago.
I think human limitation is NOT 'impossible'!

I do something http://www.mathhelpforum.com/math-he...-function.html
And why isn't e^x an elementary function ? Is it because you decided so ?

Originally Posted by Wikipedia
In mathematics, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and nth roots through composition and combinations using the four elementary operations (+ – × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by using complex variables and the relations between the trigonometric functions and the exponential and logarithm functions.

math2009 : your attempt to prove that there is a close form to the integral $\int_0^x e^{-x^2} ~dx$ is useless.
I don't know the proof, nor if there is one, but many mathematicians would have tried before you.
Approximations are nothing but approximations. It doesn't help in knowing if something has a close form. It's like you were looking for a simple form of $\pi$ or $e$.

8. It's current definition, but we couldn't solve many problems according to it.
So what do we do next steps ? Do nothing because professional mathematicans could not solve or tell you impossible.
Or change something!
If nobody challenge it, it never solve itself.