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:
- online services like isc.carma.newcastle.edu.au or wolframalpha.com,
- or my own program that performs search through a large space of closed-form expressions.
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
- oeis.org, inluding its superseeker service,
- or Mathematica commands 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?