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There is a popular question on this site which is similar to this but doesn't answer this question. How can it be proven that $\int e^{-x^2}dx$ does not exist in terms of elementary functions?

Context. The function $f(x) = e^{-x^2}$ is one of the most famous and important functions in Mathematics. It is commonly called the gaussian function, although by reading the wikipedia article on this function, the form $e^{-x^2}$ seems to be a particular case of a more general class of "gaussian functions".

As it's know, the gaussian function as it is shown here occurs everywhere in Probability Theory, and because Statistics is spoken in the language of probability, the gaussian function appears everywhere in Statistics as well, particularly in Mathematical Statistics (what we call Statistical Inference) and in Multivariate Statistical Analysis. In particular, in probability theory the gaussian function shows up in the PDF of the Normal and Multivariate Normal distributions.

If you care enough about investigating this function further, you will find out that it can be integrated $e^{-x^2}$ over $\mathbb{R}$ using polar coordinates, and that there exists a function, called the Error Function, which is denoted by $\operatorname{erf} x$ and defined as $\operatorname{erf}(z) = (2/\pi) \int_{0}^{z} e^{-t^2}dt $, where $z$ is a complex variable.

Issue. If you have ever taken a calculus-based probability class, you'll definitely and simply be told that

$$ \int e^{-x^2}dx $$ does not exist. Most people just accept this and don't further investigate it. In particular, what doesn't sit well with me is that this fact is never proved. Even at prestigious universities, such as Harvard, students are just told that the antiderivative does not exist. In the linked video, you have a class from Harvard professor Joe Blitzstein and he mentions in it that it has been proved that the antiderivative doesn't exist. My question is, how?

Edit. While there is no explicit answer (which is understandable), some comments were very helpful. The linked questions were

This question. This other question, and especially this paper.

Sigma
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    "Does not exist" is a little imprecise, maybe. A clearer statement would be "No elementary function is an antiderivative of $e^{-x^2}$". – aschepler Feb 26 '21 at 00:52
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    this area is called Differential Galois Theory. https://en.wikipedia.org/wiki/Differential_Galois_theory – Will Jagy Feb 26 '21 at 00:53
  • I'm not previously familiar with it, but the https://en.wikipedia.org/wiki/Risch_algorithm looks like a good starting point for more research. – aschepler Feb 26 '21 at 00:59
  • See also https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) in this area. The concept of "elementary function" is interesting from the classroom point of view (it corresponds roughly to what most people see/build from examples in calculus textbooks), and from the point of view of the concepts that it led to, and so on, but for many real-world purposes (e.g. ODE, PDE) it is, if not entirely artificial, often unnatural. So the fact that something doesn't have an elementary antiderivative, outside of the context of these areas, is a big "so what" to many – leslie townes Feb 26 '21 at 01:04
  • @WillJagy interesting, although it seems very sophisticated. – Sigma Feb 26 '21 at 01:10
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    It is sophisticated, Sigma. Simple-seeming questions often have complicated answers. See Theorem, Fermat's Last. For $\int e^{-x^2}$, see https://math.stackexchange.com/questions/1877889/can-the-risch-algorithm-actually-prove-that-e-x2-has-no-closed-form-antide – Gerry Myerson Feb 26 '21 at 01:12
  • @GerryMyerson yes, I’m familiar with fermat’s last Theorem. Do you know if there exists a paper or passage in a textbook of this field that contains a proof of the non existence of the anti derivative? – Sigma Feb 26 '21 at 01:20
  • This function could be integrated by using polar coordinates. – Obsessive Integer Feb 26 '21 at 01:20
  • Did you look at the link that I gave? See also http://math.stanford.edu/~conrad/papers/elemint.pdf – Gerry Myerson Feb 26 '21 at 01:24
  • @ObsessiveInteger yes, I mentioned that in the body of the question. The question is about the non existence of the anti derivative. – Sigma Feb 26 '21 at 01:25
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    @Obs, no, it can't. You can find $\int_{-\infty}^{\infty}$ using polar coords, but you can't find an indefinite integral for this function that way. – Gerry Myerson Feb 26 '21 at 01:25
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    I wish you would stop writing "non-existence" already, Sigma. You have been told that the antiderivative exists; it just can't be expressed in terms of the familiar functions of 1st-year Calculus. – Gerry Myerson Feb 26 '21 at 01:27
  • @GerryMyerson yes, I looked at the link you gave. I'll try to read the article you linked this weekend, thanks. About me writing "non existence", I wrote it due to force of habit - I wrote what is commonly told. I know that it just can't be expressed in terms of "elementary" functions, I mentioned this on the body of the question. – Sigma Feb 26 '21 at 01:31
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    Try to get out of the habit – it's a bad one. And the next time people try to tell you it doesn't exist, correct them. – Gerry Myerson Feb 26 '21 at 01:37
  • @GerryMyerson Quintics generally cannot be solved with finite term of radicals. It is as if the word radical has been replaced by the elementary function word here. The quintic equation has also been replaced by the integral. I know I did philosophy, but I felt something like that.. – lone student Feb 26 '21 at 01:47
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    @Achilles, what's an example of a (continuously differentiable) function easily proved to be non-elementary? Once you have one of those, its derivative would be a continuous function for which an easy proof of non-integrability in elementary functions could be given. – Gerry Myerson Feb 26 '21 at 02:01
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    @lone, yes, the analogy between polynomials in radicals and antiderivatives in elementary functions gives rise to the Differential Galois Theory that Will Jagy mentioned way upthread. – Gerry Myerson Feb 26 '21 at 02:03
  • @GerryMyerson Thank you for answer. – lone student Feb 26 '21 at 02:07

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