Is there a proof that there is no elementary integral for e^-x^2?

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Discussion Overview

The discussion revolves around the question of whether there is a proof that the integral of e^-x^2 is not elementary. Participants explore potential methods for proving this, including references to differential Galois theory and Liouville's Theorem.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about a proof for the non-elementary integral of e^-x^2.
  • Another suggests that differential Galois theory may be the best approach to proving the non-elementary nature of the integral.
  • A participant expresses uncertainty about how Galois theory could be applied in this context and requests further clarification.
  • There is acknowledgment of limited knowledge regarding Galois theory among participants.
  • One participant mentions finding a potential method using Liouville's Theorem, despite being unfamiliar with it.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to prove the non-elementary nature of the integral, and multiple competing views and uncertainties remain regarding the applicability of different theories.

Contextual Notes

Limitations include varying levels of familiarity with Galois theory and Liouville's Theorem among participants, which may affect the depth of discussion and understanding of the proposed methods.

Pi3.1415
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Hello, I was just wondering if anyone new of a proof that their is no elementary integral for: e^-x^2
Any help would be appreciated. :smile:
 
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Pi3.1415 said:
Hello, I was just wondering if anyone new of a proof that their is no elementary integral for: e^-x^2
Any help would be appreciated. :smile:

Hi Pi,

If you're interested in proving these, it looks like differential galois theory is your best bet. If you just started taking calculus though, you might have to wait a bit to prove it. :redface:
 
gb7nash said:
Hi Pi,

If you're interested in proving these, it looks like differential galois theory is your best bet. If you just started taking calculus though, you might have to wait a bit to prove it. :redface:

Thank you. My knowledge of Galois Theory is weak to put it in a good light, but i don't see how it could be applied in this situation. Could you please help?
 
Pi3.1415 said:
Thank you. My knowledge of Galois Theory is weak to put it in a good light, but i don't see how it could be applied in this situation. Could you please help?

Unfortunately, my knowledge on galois theory is very limited. There might some other simpler method but I don't know it.
 
Okay thank you anyway; I *think* I've found a way using Liouville Theorem (which I've never heard of before).
 

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