0

The Wikipedia article on Risch's algorithm makes this interesting statement:

the following algebraic function has an elementary antiderivative: $$f(x) = \frac{x}{\sqrt{x^4 + 10 x^2 - 96 x - 71}}$$ [...] But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions.

For the first part of the statement the antiderivative is given, together with a (quite cool) reference to a post by the late Bronstein. But for the second part of the statement no reference is given (as often happens in Wikipedia; so the second part is just a claim). I tried to check the book by Geddes & al but didn't manage to find that statement there.

Does anyone know any reference to a proof that if 71 is changed to 72 no antiderivative exists that can be expressed in terms of elementary functions?

pglpm
  • 831
  • Isn't it just crank it in Risch algorithm to decide? Granted the proof that the algorithm works is quite long and splits into algebraic and transcendence cases, with the algebraic case being harder. – user10354138 Feb 10 '21 at 17:38
  • @user10354138 I don't know if it's "just". I'm not familiar with the algorithm. I suppose many Wikipedia readers who reach that statement aren't thoroughly familiar with it either. – pglpm Feb 10 '21 at 17:58
  • 1
    Then you should actually look at the (over 100 pages) description of Risch algorithm in Geddes et al (1992) in the references section of the wikipedia article, or the Bronstein (1998) (also in Reference) for a more concise outline. – user10354138 Feb 10 '21 at 18:21
  • @user10354138 Thank you, indeed I managed to find Geddes & al – checked there for a proof on Wikipedia's claim but couldn't find anything. I'll read it thoroughly as soon as I have more time – looking forward to it. But in the meantime I'd really like to know if there's a proof of that claim. Can't just take for true something because it's written on Wikipedia. – pglpm Feb 10 '21 at 18:33

2 Answers2

1

Comment:May be this idea helps:

$x^4+10x^2-96x-71=(x^2+5)^2-96(x+1)=$

Let:

$x^2+5=u\Rightarrow 2xdx=du\Rightarrow dx\frac {du}{2x}=\frac{du}{2\sqrt{u-5}}$

Putting this in integrand you get:

$$ \frac{dx}{x^4+10x^2-96x-71}=\frac{du}{[u-96(\sqrt{u-5}+1)](2\sqrt{u-5})}$$

Which can be transformed to two fractions.This is not possible if you replace 71 by 72.

sirous
  • 10,751
  • Thank you sirous. Yet this is not a proof that for the expression with "72" it is impossible to represent the antiderivative in terms of elementary functions. One wonders if it might be possible by using some different idea – pglpm Feb 10 '21 at 16:28
  • @pglpm, Just wanted to give an idea, it is certainly not a proof. – sirous Feb 10 '21 at 16:30
  • This is useful for the first part of the statement indeed. You could add this on the Wikipedia article, it'd be very informative. – pglpm Feb 10 '21 at 16:31
1

This integral was solved by Chebyshev here https://archive.org/details/117744684_001/page/n11/mode/2up

Ultimately because fractional expansion of this function in denominator has a specific form we know integral is elementary and we construct a subtitution from expansion coeff. See https://math.stackexchange.com/a/3933268/1013030

What is funny full proof for that was only given by Zolotarev. enter image description here

Valentin
  • 46
  • 2
  • That's great, thank you. The problem here was not the proof of the existence of an analytic form when the coefficient is 71, but the proof of the impossibility of an analytic form when the coefficient is 72. These papers give the proof for this latter part. They should be referenced in the Wikipedia page, if they aren't already. – pglpm Feb 03 '22 at 11:01
  • 1
    I will add them to wikipedia, fine. And yes, I did mentioned that we know WHEN (iff condition) the antidervative is elementary. – Valentin Feb 03 '22 at 13:37
  • 1
    Added. BTW, https://en.wikipedia.org/wiki/Yegor_Ivanovich_Zolotarev outright talks about this integral, ha. – Valentin Feb 03 '22 at 14:00