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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?

Thanks in advance.

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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
up vote 1 down vote accepted

By can't expressed in closed form form one usually means that it is not an elementary function.

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

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Is the same true for elliptic integrals as well? – dexter04 Dec 17 '12 at 16:41
They are not elementary, but I do not know if this follows from Liuoville's theorem. – Julián Aguirre Dec 17 '12 at 16:43
K. I am more interested in the elliptic case. But thanks for the pointer to Liouville's theorem – dexter04 Dec 17 '12 at 16:46
@dexter, then you might find this interesting... – J. M. Apr 3 '13 at 12:52

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