I am writing a physics paper, and I have decided not to write down explicit equations I have found since they are incredibly long. Instead I just report plots showing the behaviour resulting from these equations.

Is it deprecable to say something like the following?

We have found analytic results, but we do not report them here for sake of brevity.

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    considered including the equations as supplementary material?
    – posdef
    Commented Jul 25, 2014 at 11:02
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    "Please refer for the analytic results to Appendix~\ref{foo}." If you are not presenting the equations (imho this is also true for experimental data) at all, it looks like you are hiding something. If you have to move the appendix to your website later, because a journal has restrictions on the length of the paper it is fine. Commented Jul 25, 2014 at 11:14

4 Answers 4


Idealistically, a paper should aspire to contain all the information required for reproducing its empirical results and verifying its deductive results with reasonable effort. Just writing the suggested sentence, however, does not allow for this, as someone would have to redo your work on finding those equations from scratch.

This may drastically reduce the usefulness of your paper, as it unnecessarliy increases the amount of work other people have to put in using your results and decreases their perceived soundness (see Fermat’s Last Theorem for an extreme example). This may also be harmful from an egoistical point of view as it does not improve your popularity amongst others in your field and you may receive less citations.

Depending on how exactly your equations were derived and look like, the following ways to include or not include them may be appropriate:

  • From [equations derived in the paper] we can obtain closed solutions for [variables] using [standard technique or computer algebra system], which we use in the following.

  • From [equations derived in the paper], we obtain closed solutions for [variables] (see Appendix X).

  • If the target journal does not allow for appendices and has a content limit (in which case it will usually be a letter journal), the following may be acceptable:

    From [equations derived in the paper] we derived analytical results for [variables]. For brevity’s sake, these results are not given here and will be published elsewhere.

    Something similar may also be appropriate if you are publishing in a journal of another discipline, e.g., you are publishing in a medical journal as your equations are relevant for an imaging technique.

    Either way, it should go without saying that publishing the results elsewhere should be a realistic endeavour and actually be intended. Also, if you are not giving too much away, it would be better in terms of soundness and acceptance to actually publish your analytical work first.

As always, you will likely get a better answer from someone familiar with your work and field, such as your supervisor.

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    I would emphasize that if a paper is missing information needed to reproduce its results, it can have a serious negative impact on other people's research process. I've been dealing with exactly this problem for a while, trying to manipulate some results of a series of theoretical papers where the derivations are not given in enough detail to allow me to reproduce them.
    – David Z
    Commented Jul 25, 2014 at 12:14
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    Replace "will be published elsewhere" with "are available in the arXiv version of this paper [##]".
    – JeffE
    Commented Jul 25, 2014 at 12:50
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    @JeffE: If publishing on the ArXiv only is an accepted way in the respective field, this is only a slight variation of “will be published elsewhere”. If not, this may not be well-received by the journal or the reviewer’s. (Also, do not forget that for some journals the best you can do is publish the derivations on their own, as they do not allow ArXiv publication.)
    – Wrzlprmft
    Commented Jul 25, 2014 at 13:01
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    @Wrzlprmft My reaction as a reviewer to "will be published elsewhere" is "Suuuure it will. Reject." If the missing details are on the arXiv (totally standard in physics), then they're not really missing, so I am more willing to trust the results.
    – JeffE
    Commented Jul 25, 2014 at 13:06
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    And if the journal doesn't allow arXiv publication, choose a different journal.
    – JeffE
    Commented Jul 25, 2014 at 13:07

You might want to consider attaching a Mathematica notebook to the arxiv submission, which is something I've been seeing relatively often lately.


Can you add an appendix to the paper? If not, I think whether it is reasonable to add the statement you proposed depends on how central the analytical results are to the findings in your paper.

If the result isn't important, I think what you propose is fine.

If the analytical result is crucial, then you need to think very carefully about making the result available to a reader -- maybe an extended version of the manuscript on arxiv? Or on your website?

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    The result is indeed crucial. The fact is that the results are unusually long, we manipulated them through the use of Mathematica. I think is impossible to write them even on several pages... Commented Jul 25, 2014 at 11:42
  • In my field (Neuroscience), it it possible to make data or analysis software available with a publication (usually as supplementary material). You could try to take that approach -- write to the editors and ask them whether supplying supporting code is an option for you. Commented Jul 25, 2014 at 11:53
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    I strongly disagree with "If the result isn't important, I think what you propose is fine." How do you know that the result will never be important? Some papers are frequently cited for something that the authors though at that time was a minor comment. I think if the result isn't important you either do not publish it or make it "verifiable". Commented Jul 25, 2014 at 14:13
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    @giulio: You might want to edit the question to mention that the equations were generated using Mathematica, since it could make a difference in the answer. For instance, you could consider including the Mathematica code in your paper, rather than its output. That might be shorter, and easier for a reviewer to check. Commented Jul 25, 2014 at 14:24

I was in a somewhat similar position some time ago and decided against publishing the analytical results, since they were too long. Essentially, I was solving a linear system of equations with Mathematica, which can produce horribly long equations, which are hard to analyze any further. Also, computers might actually be faster in solving the initial linear system numerically, then evaluating long arithmetic equations. For these reasons, I did not stress that I could have written down the solution in analytical terms in the paper. I just gave the linear system and said that we produced the plot from its solutions.

In summary, I doubt that the result is crucial, when the analytical equations are several pages long. That being said, if you can make the equations available (preferably in some electronic version, since nobody is gonna type them into a computer), that is certainly a good solution.

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