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The Risch algorithm is used to find closed-form antiderivatives.

If I understand the article right, only heuristics are known. On the other hand, I came across the claim that it is known that $e^{-x^2}$ has no closed-form antiderivative.

Is the Risch-algorithm successful in the case $f(x)=e^{-x^2}$ ? Is it actually proven that no closed-form antiderivative exist or just very probable because no form has been found using the Risch algorithm ?

This question could well be a duplicate, but I am not sure whether the aspect of decidability has been asked.

Caleb Stanford
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Peter
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1 Answers1

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If an elementary derivative exists then the Risch algorithm will construct it. Otherwise, the proof of the correctness of the algorithm implies that no elementary antiderivative exists. The algorithm could be modified to output witnesses that no solution exists, but that is rarely done in practice since there is no need to do so.

Most expositions of the algorithm include small worked examples (such as yours) which will help you gain intuition on how the algorithm works, e.g. see the links I gave in this prior answer, esp. max Rosenlicht's Monthly expsoition, which is one of the most readable introductions. There he proves the classical Liouville criterion, that for rational functions $\,f,g\,$ the integral $\int f e^g$ is elementary $\iff$ there is a rational function $\,h\,$ such that $\,f = h' + h g'$.

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Bill Dubuque
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    My understanding was that current implementations of the Risch algorithm in computer algebra systems weren't really complete implementations of the Risch algorithm. Has this been resolved in recent versions of these packages? – Brian Borchers Aug 01 '16 at 14:32
  • Early CAS version had incomplete implementations. I completed the transcendental case of the Risch algorithm in Macsyma (incorporating some of Manuel Bronstein's ideas). I suspect that Maple and Mathematica also now have complete implementations of the transcendental cases too. The algebraic case is much more complex and was never completely implemented in Macsyma, primarily because there was no (commercial) demand for such. Most of the integrals encountered in pratice were handled by lower-dimensional special cases (e.g results of Davenport and Trager). – Bill Dubuque Aug 01 '16 at 15:09
  • The algebraic case was completely implemented by Barry Trager in Scratchpad. By now I would not be surprised if the algebraic case is also completely implemented in Maple or Mathematica. Note that the open source version of Maxima is based on a very old sources (circa 1982), and has a (very) incomplete Risch implementation. I don't know if anyone has updated it in Maxima. – Bill Dubuque Aug 01 '16 at 15:11
  • The Wikipedia article on the Risch algorithm (under the decidability heading) specifies that the Risch algorithm is not a true algorithm due to an inability to algorithmically determine if division by $0$ is occuring. How has this issue been circumvented in the actual implementations of the Risch algorithm? – Jason DeVito - on hiatus Aug 01 '16 at 15:15
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    @JasonDeVito That theoretical problem has no impact in practice. The problem is that for nontrivial constant fields that contain enough transcendental numbers (such as $,e,\pi),$ equality is undecidable in general (it leads to difficult problems in transcendental number theory), Of course if you cannot decide it a constant is zero or not, then you can't do much of anything over the field. But this has zero impact in practice. – Bill Dubuque Aug 01 '16 at 15:30
  • @Bill: That make sense. Thanks for clarifying! – Jason DeVito - on hiatus Aug 01 '16 at 15:30
  • @BrianBorchers Do you agree that the Risch-algorithm always terminates and proves that no antiderivate exists ? – Peter Aug 01 '16 at 15:54
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    Yes- my point was that actual implementations of the algorithm in CAS software aren't always complete implementations of the algorithm. There's a distinction between Risch's algorithm as published and implementations of the algorithm in particular software packages. – Brian Borchers Aug 01 '16 at 16:27
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    I specifically asked on MathOverflow whether there exists a complete implementation of the Risch algorithm in any CAS, and the answer seems to be no (as of October 2020). Also, regarding zero-recognition of constants, Richardson described an algorithm to solve that problem; the only "catch" is that we have no rigorous proof that the algorithm terminates (the termination is contingent on Schanuel's conjecture being true). But the algorithm always terminates in practice, and when it terminates, the answer it gives is correct. – Timothy Chow Oct 15 '20 at 22:03
  • @Tim You received one answer on MO claiming that (but giving no support for that claim). The constant problem has already been mentioned above. – Bill Dubuque Oct 15 '20 at 22:17
  • @BillDubuque : There is support. But you have to read his arXiv preprint, where he gives success rates (less than 100%) for various CAS. And yes, I noticed the earlier comments about the constant problem, but what you didn't mention is that for "elementary" constants, the problem is not undecidable, but is in fact decidable as long as you believe Schanuel's conjecture. See Richardson's paper on "The elementary constant problem." – Timothy Chow Oct 15 '20 at 23:39
  • @Tim Likely he mentions only commercial systems. A lot of the implementations never made it into commercial systems. Barry Trager was a fellow member of the Macsyma group when we were both students at MIT so I am familiar with what he did there and at IBM research (ScratchPad). If I remember correctly he did a complete implementation of the classic Risch algorithm (with some improvements). The results you mention about the constant problem are well known and have been mentioned here before, e.g. here. All of these results are many decades old. – Bill Dubuque Oct 16 '20 at 00:39
  • @BillDubuque: Are you unable to access the arXiv and see for yourself? No, he doesn't mention only commercial systems. E.g., he mentions FriCAS, which is based on Axiom and so likely contains whatever was in ScratchPad. I don't know for sure what Trager did, but his thesis doesn't consider the mixed algebraic-transcendental case. I'd be surprised if he did more than what Bronstein did, and Bronstein admitted his implementation was incomplete. And obviously I know Richardson's paper is old but you didn't mention it in the current thread so I thought it was worth pointing it out to Jason. – Timothy Chow Oct 16 '20 at 00:49
  • @Tim The internet is full of nonsense. Be careful what you choose to believe. The algorithms have been known for many decades. – Bill Dubuque Oct 16 '20 at 01:01
  • @BillDubuque : Yes, I know that the algorithms have been known for decades. That's not the question. I am following your advice and choosing not to believe what you say unless you can provide more evidence. – Timothy Chow Oct 16 '20 at 01:03
  • @BillDubuque : I emailed Barry Trager and he responded promptly, confirming that he worked on the purely algebraic case, and Bronstein extended it to the mixed algebraic-transcendental case, but it was not a complete implementation. He said he was not aware of any complete implementations but wasn't sure. – Timothy Chow Oct 16 '20 at 13:30
  • @Tim Looking closer it seems the misunderstanding above concerns the denotation of "the (classic) Risch algorithm", which wasn't clear based on your initial comment above. – Bill Dubuque Oct 16 '20 at 22:35