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It often happens that people conjecture possible closed forms of integrals, series, and so on starting from a numerical value calculated to very high precision.

What are the techniques, tricks, methods, and softwares (in particular, what are the function of Mathematica and Maple) that help in formulating such conjectures?

Any reference is highly appreciated.

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    $\begingroup$ en.wikipedia.org/wiki/Inverse_Symbolic_Calculator $\endgroup$
    – Will Jagy
    Commented Oct 13, 2014 at 20:22
  • $\begingroup$ Note that I've added a bounty because the question has been edited to have a broader scope (compared to when it was first answered). $\endgroup$
    – Dal
    Commented Oct 20, 2014 at 11:32
  • $\begingroup$ I don't know whether this technique is not meant: I've often related values gotten by approximations, and the first step is then to look at the arithmetic or geometric progression and remove parts of the numbers to come nearer to the common core. This can help much to enter the correct path to go. $\endgroup$ Commented Oct 26, 2014 at 4:37

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Here are a few books as stated in this answer:

The first two books reference each other, so I recommend having both on hand. The third is in my reading queue. All are by Jonathan Borwein and David Bailey.
Experimentation in Mathematics
Mathematics by Experiment
Experimental Mathematics in Action

These books cover Mathematica, Maple, and many other tools.

Edit After reading the third book, I suggest you read it first.

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  • $\begingroup$ I didn't realize @Dal had posed the other question when I posted this answer. Oh, well... $\endgroup$ Commented Oct 20, 2014 at 11:56
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Need 30 characters..................

http://isc.carma.newcastle.edu.au/standard

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Is there a function in Mathematica or any other software that allows you to find closed forms for decimal numbers calculated with high precision?

  • Maple has the identify command. The Inverse Symbolic Calculator is based on Maple.

  • In Mathematica, there are Rationalize and Recognize, the latter of which requires the installation and appellation of the NumberTheory package. But they are only useful for either rational or algebraic numbers, and cannot identify transcendental expressions. Newer versions, however, do include commands like

    WolframAlpha["...", IncludePods -> "PossibleClosedForm", AppearanceElements -> {"Pods"}]
    
  • This link might also prove helpful.

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  • $\begingroup$ Thank you for your answer, but how do I install a new package (namely, NumberTheory) in Mathematica? $\endgroup$ Commented Oct 14, 2014 at 15:35
  • $\begingroup$ @mathlearner: That would depend on your specific version. In older ones, go to the Mathematica folder, and check for ...\AddOns\StandardPackages\NumberTheory\.... If it's already there, and it contains the files Rationalize.m and Recognize.m, then all you have to do is appellate it from the program $($just go to Help $\to$ Master Index, and type in Recognize: you'll find there all the necessary information$)$. If not, then it has to be copied there from another location, which is also found inside the Mathematica folder. $\endgroup$
    – Lucian
    Commented Oct 14, 2014 at 15:58
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One of the oldest ISC (Inverse Symbolic Computer) is the Plouffe's inverter, cited here for memory because it is no longer available at http://pi.lacim.uqam.ca/

Others were already quoted in the peceeding answers : http://oldweb.cecm.sfu.ca/ http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html

Also, WolframAlpha provides such tool : https://mathematica.stackexchange.com/questions/55818/lookup-in-inverse-symbolic-calculator-from-mathematica

I am rising here to mention that, surprisingly, it is not very difficult to built an "homemade" ISC on a personal computer. Of course, it will be much less effective than the professionnal ISC cited above. Doing such a software by ourself is more an educational game to experiment some elementary features of those kind of tools. There is a short general-public paper publish on Scribd which gives an hint on the subjet (in French, not translated yet) : http://fr.scribd.com/doc/14161596/Mathematiques-experimentales

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