# Closed form solution of elliptic or gaussian functions

I have often read that integrals of functions like $\exp(-\frac{x^2}{2})$ or $\frac{1}{\sqrt{1-k^2\sin^2\theta }}$ have no closed form solutions. I am unable to find what closed form exactly means, though I get the rough idea that it is polynomials, trigonometric ratios, exponents, their compositions and inverses.

So, here are my two doubts:

1.Is there a proof that the integrals can't be expressed in closed from (whatever definition you assume), or is it that nobody has found them out?

2.What about functions like Bessel functions? Do they count as closed form?

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Some informations and links here. Bessel functions are usually not considered as closed form. – Raymond Manzoni Dec 17 '12 at 16:41
@Raymond, "Bessel functions are usually not considered as closed form." - that most certainly depends on who you ask... – J. M. Apr 3 '13 at 12:48
Yes @J.M. of course ! :-) (Glad to have you back by the way !). I should have added 'in terms of elementary functions' here. – Raymond Manzoni Apr 3 '13 at 14:37

Liouville's theorem of differential algebra proves that certain antiderivatives (of $e^{-x^2}$ for instance) are not elementary functions.