My hobby is to seach for closed forms of integrals that cannot be evaluated by modern CAS like Maple or Mathematica.

Sometimes I can intuitively find the right way to transform the integral and can get the result with a rigorous proof. More often, I start from guessing a closed form using different tools:

If an integral is parameterized, I iterate through several fixed values of a parameter (often, integers) and try to guess a closed form for each of them. If this approach succeeds, I can get a list of conjectured values, that may show an obvious pattern. More often though, the pattern is non-obvious, but can be discovered using tools like

Then I can assume that the found expression is valid not only at integer values it originated from, but for arbitrary complex or real values in a certain interval, and it also can be differentiated or integrated w.r.t the parameter (when the original integrand is also transformed in a corresponding way). Sometimes numerical checks reject this assumption, but they also may support it with hundrends or throusands digits of precision.

In the latter case I try to find a rigorous proof of the found conjectural closed form (because, obviously, I obtained it with completely heuristic and non-rigorous methods). But "knowing" the result is often helpful when looking for a right approach to prove it.

Sometimes all my attempts to prove the identity failed, but all numerical methods still do strongly support it, and I wish to share my conjecture with other people, in hope they could prove it or provide some useful ideas. I feel that math.stackexchange.com is a good place to post my conjectures. I have done it several times, but every time I felt somewhat uneasy, felt that I owe some explanations. But I cannot possibly post dozens of pages of work that led me to the final conjecture every time.

I am asking for your advice, what is the best way to post my conjectures, without saying too much words for introductions, but to attract people's attention and make the problem interesing for them?

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    $\begingroup$ For me it would be enough if you just posted a formula, stated it was a conjecture, and asked for help. If it within my area of interests and expertise, I would try to prove it. $\endgroup$ – Oksana Gimmel Oct 5 '13 at 18:19
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    $\begingroup$ In addition to the suggestions by Oksana, I suggest to add a brief account of what you tried. Some partial results, numerical approaches, generalisations thereof that failed, approaches that failed. This prevents precious duplication of work and shows that you care about the problem. It may also provide some leads for others to follow up on. I feel that questions phrased that way will have an excellent reception on MSE. $\endgroup$ – Lord_Farin Oct 5 '13 at 19:45
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    $\begingroup$ In my opinion, m.se is not for posting conjectures. But you could post a reference request. "I have heuristic evidence that $\int\cos x\,dx=\sin x+C$; does anyone know a reference in the literature to this integral?" $\endgroup$ – Gerry Myerson Oct 6 '13 at 0:48
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    $\begingroup$ Your questions are always interesting, by the way. You may consider adding a comment at the end, pointing to this question, so people may get some additional context, if you feel it may be relevant, but too distracting to include in the body of the question. $\endgroup$ – Andrés E. Caicedo Oct 6 '13 at 6:52
  • $\begingroup$ I take it the conjectures concern definite integrals rather than indefinite integrals? $\endgroup$ – hardmath Oct 6 '13 at 20:07
  • $\begingroup$ (Examples of potentially relevant context: Has the accuracy of your computations been verified? How? Are there similar closed form expressions in the literature? If so, why are the techniques used to established those not sufficient here?) $\endgroup$ – Andrés E. Caicedo Oct 6 '13 at 21:27

You have a conjecture. You think it might be true, and you think it's interesting. Convince the reader of these two points. Just look ask yourself why you think it's true, and why you find it interesting, then write those reasons out in your post.

The worry is that this will lead to an overly long question. Follow the inverted pyramid rule. A reader who's just interested in the technical details should be able to get thoes within the first few lines and skip the rest. Motivation should be written further down, for readers who are interested.

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    $\begingroup$ I am not sure about the rule. The technical details at the beginning may put off people who might have otherwise approached the problem through other angles. $\endgroup$ – Andrés E. Caicedo Oct 6 '13 at 22:06
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    $\begingroup$ @AndresCaicedo Ideally there should be a bit of motivation before the technical details, but not a whole essay. I tend to leave if a questioner seems to just be waffling on - some questions contain an awful lot of description and it's really very hard to see what's being asked. $\endgroup$ – Jack M Oct 6 '13 at 22:11
  • $\begingroup$ @AndresCaicedo, I think there's two kinds of motivation. There's the "how I stumbled on the conjecture and why I think it might be true" kind of motivation, and then there's the "supposing its true, how would that be useful" kind of motivation. I think its a good idea to put the first kind of motivation at the beginning, ask the question, and then put a longer discussion that may include the second kind of motivation at the end. So, I think the technical details should go somewhere in the middle. $\endgroup$ – goblin Oct 11 '13 at 7:34

Just post the conjecture, along with some commentary documenting the numerical evidence that led to it, and any proof attempts you have tried that didn't succeed.

I think such a question is perfectly appropriate for math.se. Those who don't appreciate such open-ended question can always ignore the conjectures tag.


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