Software, techniques and tricks of experimental mathematics to conjecture possible closed forms

Question

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.

Answer 1

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.

Answer 2

Need 30 characters..................

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

Answer 3

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.

Answer 4

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