What makes a non-elementary integral non-elementary?

Asked Dec 11 '22 at 15:37

Active Dec 11 '22 at 15:46

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I was looking over the popular non-elementary integrals like the gaussian and fresnels, and I saw that there were some patterns. They all of course do share the trait of being impossible to express algebraically, or trigonometrically, logarithmically, etc.

So I'm curious: is there another specific trait of these functions that they all share that makes them impossible to integrate?

edited Dec 11 '22 at 15:46

Peter

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asked Dec 11 '22 at 15:37

Mathewpotao

3

This question is probably too vague to give a satisfying answer. – jjagmath Dec 11 '22 at 15:44
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The answer is essentially the Risch algorithm, though it's not especially intuitive. – Ian Dec 11 '22 at 15:45
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What you are looking for is the topic of differential algebra. See for example this link. – Lee Mosher Dec 11 '22 at 15:46
@jjagmath perhaps to someone more versed in mathematics it seems to be so, but to me, the question is simple: I see this rule that all functions we can't integrate abide by. Are there others? – Mathewpotao Dec 11 '22 at 15:46
@ian alright I'll check it out thank you – Mathewpotao Dec 11 '22 at 15:46
@LeeMosher ah ok, thanks a bunch! – Mathewpotao Dec 11 '22 at 15:47
Integrating is much harder than derivating and we soon arrive at expressions that have no closed form antiderivate. Even a product of two functions with antiderivates is too much in general. – Peter Dec 11 '22 at 15:48
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@Mathewpotao "I see this rule that all functions we can't integrate abide by." Which rule do you see? I don't see a similarity between $x^x$ and $\cos(\cos(x))$ – jjagmath Dec 11 '22 at 15:52
@jjagmath I explained that in the original comment. I was curious as to what other properties these functions shared, if any. – Mathewpotao Dec 11 '22 at 15:53

What makes a non-elementary integral non-elementary?

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