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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](http://math.stackexchange.com/questions/239105/special-integrals). Bessel functions are usually not considered as closed form.2012-12-17
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    @Raymond, "Bessel functions are usually not considered as closed form." - [that most certainly depends on who you ask...](http://math.stackexchange.com/a/9203/498)2013-04-03
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    Yes @J.M. of course ! :-) (Glad to have you back by the way !). I should have added 'in terms of elementary functions' here.2013-04-03

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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?2012-12-17
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    They are not elementary, but I do not know if this follows from Liuoville's theorem.2012-12-17
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    K. I am more interested in the elliptic case. But thanks for the pointer to Liouville's theorem2012-12-17
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    @dexter, then you might find [this](http://math.stackexchange.com/questions/15750) interesting...2013-04-03