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i was computing integrals, and was thinking of an anti-derivative of $\int e^{-x^2} dx $. See, i know for a fact that this function has no elementary anti-derivative, but i was curious and tried around a bit. For this i used the known fact, that $e^x=\sum\limits_{n=0}^{\infty} \frac{x^n}{n!}$, so i replaced $x$ by $-x^2$. This gives the following : $\int e^{-x^2}dx=\int \sum\limits_{n=0}^{\infty} \frac{-x^{2^n}}{n!} dx$. Now, if integrated, term by term, wouldnt this be a countable sum of elementary functions, thus making it elemetary? I would appreciate if anyone could point out my flaw(s) here. Maybe i just cant substitute $x$ for $-x^2$?

bsvgu
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  • "Elementary" is just a definition. You shouldn't worry too much about it. You'll probably find that your antiderivative is the so-called "error function". – Benjamin Wang Jan 04 '21 at 17:18
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    Why do you say that a countable sum of elementary functions is elementary? That's not generally true with the usual definitions - only finite sums are explicitly allowed. – Matthew Towers Jan 04 '21 at 17:20
  • I just assumed countable infinite sums would also count, since $e^x$ is also defined as one. – bsvgu Jan 04 '21 at 17:22
  • Do you also believe a convergent infinite sum of rational numbers must be rational? For example, $1 -1/3 + 1/5 - 1/7 +$... ? – SenZen Jan 04 '21 at 17:33
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    No, i know this to be irrational, but how does this exactly proof an infinite sum of polynomials to be non-elementary? – bsvgu Jan 04 '21 at 17:39
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    @bsvgu infinite sums of polynomials can be elementary, like the sum defining $e^x$ for example. But they're not guaranteed to be elementary. Deciding whether or not a given power series is elementary or not is hard, that's why proving $\int e^{-x^2} dx$ isn't elementary is difficult. – Matthew Towers Jan 04 '21 at 18:17

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I should note that its actually $(-x^2)^{n} = (-1)^{n}x^{2n}$.

There's no problem substituting $x$, in fact it helps define the error-function.

Anyway, the function you can get from this substitution is not elementary, simply because it's an infinite sum of polynomials!

See the definition of an Elementary Function here.

Edit: Thank you Matthew Towers for correcting me - yes, an infinite polynomial can be an elementary function - for example, $\sum_{n \geqslant 0} \frac{x^n}{n!}$. Although some infinite polynomials are elementary functions - some aren't, for example your integral (or the integral of $e^{x^2}, x^x$ and a bunch more). Proving this isn't very easy - it's actually quite difficult.

Unfortunately, as much as I hate saying this (especially in mathematics), you just have to agree that the integral you presented is not elementary, since proving this requires a lot of extra knowledge - which unfortunately I don't know. If you still want to read some more about this, here are a few more questions on Math SE which talk about this:

  1. How can you prove that a function has no closed form integral?
  2. Why can't erf be expressed in terms of elementary functions?
  3. Proving that special functions do not have closed-form expression
  4. Integration of $e^{-x^2}$
talbi
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