I discovered a new analytical integral. How should I present this knowledge?

Question

During my research in theoretical physics I encountered a problem that required integrating a space of functions that I don't think has ever been done before. I discovered that these integrals are analytic and can be represented by a linear combination of elliptic and non-elliptic functions which I believe is a rather non-trivial result.

I think this research, while rather esoteric, is nonetheless important enough that the world should know. In my paper this result is buried beneath other more physics-oriented results. My question is this: how should I present this pure-mathematical knowledge to the world? Is there a journal that this kind of result would be suitable for or some mathematical database of known analytical integrals? Not being a pure mathematician I am unfamiliar with the correct protocol.

Thank you!

Edit: After discussing with a pure mathematician I decided to have another look thought Gradshteyn & Ryzhik and I think the result I obtained is probably just a linear combination of known integrals, albeit in a hidden form.

Answer 1

You actually have three choices: physics, pure math, and applied math. It might be useful in any of those domains.

But there are two questions to ask, and they are related.

In which domain is this result most useful?

How is it connected to other things within each of those domains?

A pure mathematician will find the result most interesting if you have strong connections. A physicist or applied mathematician will find it most interesting if it is useful outside its original application.

That's not to say, of course, that any of them would find it uninteresting. But they are all looking for things that can be used to extend their own work.

So, I would try to figure out where it fits most naturally and write it up accordingly, focusing on either the connections or the usefulness. Then send it to any good journal in that domain and listen to the feedback you get. Reviewers may agree that you have chosen correctly or not.

Answer 2

Please don't take this personally and keep in mind that your description is vague enough that I have to extrapolate some details and might go off in the wrong direction.

But to be blunt, I disagree that it is non-trivial. From the viewpoint of research mathematics, taken on its own, being able to integrate a certain class of functions isn't interesting in any way, especially if as you say it turns out to just be a linear combination of known results. In fact I could out dozens of such classes using the reverse process of just differentiating random terms and then reversing the calculation. So even if you would find a journal to publish this in, no one would read it.

What matters is context. Mathematicians are generally interested in the greater whole. Two possibilities come to mind in this context, but I fear neither of them is true here.

1. The proof uses interesting and novel techniques which ideally can be used in lots of different problems. Keep in mind though that novel in this context will not mean a tricky looking substitution (that's 18th century stuff), but some application of deeper theorems or other tricks usually from an entirely different area of mathematics. As a rule of thumb, if your proof is short (i.e. less than say 10 pages) and not heavily relying on citations from somewhere else, this is unlikely.

2. The class of integrals is in itself interesting. This usually means that it occurs in a lot of completely different places, which would indicate that those places have something in common that has not been explored yet. But even in this context, the solution of the integrals would not be the important part, but rather an analysis of the fact that these integrals seem to occur everywhere.

As a final remark, don't be discouraged, physicists and mathematicians just think differently and as a result have different concepts of what is interesting. I've seen the converse happen as well, where I was able to prove something about a physical problem that looked interesting to me but where the physicists I talked to did not see the point.

If you still want to continue with this, the options I see is to either just post your integral on math.stackexchange to see if someone else has seen something similar or if you want to publish your physics-result anyway, just mention your integrals in the abstract and cross-list the arxiv pre-print to math.CA (which I think should be the most fitting category).