The simplification of results of integration to logs or to arctan can depend on how the programmer viewed the importance of avoiding complex numbers.
This is not inherent in computer algebra systems generally, or even Sage particularly. It is hardly an example where a computer could not do what a human could do, My advice is you should never get into a situation where a limitation of a particular computer program --- or even several computer programs -- leads you to say that computers cannot compute something "the way humans can". Recall that for many years people claimed that computers could never play a good game of chess. On Friday, October 31, 2014 10:06:02 AM UTC-7, Christophe Bal wrote: > > > Thanks for this hint. If I find revelant example, I will post them in this > discussion. > > 2014-10-31 17:38 GMT+01:00 Aaron Meurer <[email protected] <javascript:>>: > >> On Fri, Oct 31, 2014 at 6:37 AM, Christophe Bal <[email protected] >> <javascript:>> wrote: >> > Thanks for the answers. >> > >> > Maybe integration will be a good candidate. When I talk about formal >> output, >> > I think of special case where the formula given is correct but certainl >> not >> > the one a human will see first. >> >> I see. I was thinking more of cases where the human wouldn't see how >> to solve it at all. >> >> A possible idea here is integrals using square roots. Humans tend to >> make simplifications on square roots which are not true in general, >> leading to "simpler" output, whereas computer algebra systems try to >> be more cautious about them. For instance, sqrt(1/x) != 1/sqrt(x). >> >> Aaron Meurer >> >> > >> > One day, a student had shown me an integral which was evaluate by its >> formal >> > calculator to a formula containing arctan whereas my solution do not >> use it, >> > and was more simple. The problem is that I have not noted this >> example... >> > >> > Maybe solving some polynomial of degree 3 can give such "complicated" >> > formulas that a human would not use. >> > >> > Christophe BAL >> > >> > 2014-10-30 23:53 GMT+01:00 Richard Fateman <[email protected] >> <javascript:>>: >> >> >> >> There are even simpler examples. For instance, some systems multiply >> >> polynomials by some evaluation/interpolation scheme in finite fields. >> Or >> >> by FFT >> >> or by so-called Karatsuba or Cooke-Toom methods or a Kronecker method >> >> evaluating a polynomial to a single huge integer... >> >> >> >> Risch integration is certainly another one. >> >> >> >> >> >> On Wednesday, October 29, 2014 3:40:58 AM UTC-7, Christophe Bal wrote: >> >>> >> >>> Hello. >> >>> >> >>> I'm looking for formal outputs that are not calculated as a human can >> do >> >>> (this is for a free french "book"). Do you know such kind of examples >> ? >> >>> >> >>> Christophe BAL >> >>> >> >>> PS : this questions has been posted on both Sage and Sympy lists. >> >> >> >> -- >> >> You received this message because you are subscribed to the Google >> Groups >> >> "sympy" group. >> >> To unsubscribe from this group and stop receiving emails from it, send >> an >> >> email to [email protected] <javascript:>. >> >> To post to this group, send email to [email protected] >> <javascript:>. >> >> Visit this group at http://groups.google.com/group/sympy. >> >> To view this discussion on the web visit >> >> >> https://groups.google.com/d/msgid/sympy/a2c48c91-3bc3-4761-bef7-7a8bdebc7297%40googlegroups.com >> . >> >> >> >> For more options, visit https://groups.google.com/d/optout. >> > >> > >> > -- >> > You received this message because you are subscribed to the Google >> Groups >> > "sympy" group. >> > To unsubscribe from this group and stop receiving emails from it, send >> an >> > email to [email protected] <javascript:>. >> > To post to this group, send email to [email protected] >> <javascript:>. >> > Visit this group at http://groups.google.com/group/sympy. >> > To view this discussion on the web visit >> > >> https://groups.google.com/d/msgid/sympy/CAAb4jGmGUqW1vGBkYLieOYhBGJT%3DOn7nfZDyHVNFW524%3DGfN7w%40mail.gmail.com >> . >> > >> > For more options, visit https://groups.google.com/d/optout. >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected] <javascript:>. >> To post to this group, send email to [email protected] <javascript:> >> . >> Visit this group at http://groups.google.com/group/sympy. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CAKgW%3D6JH_tCkVtKsmFxcJ4Ucbajc8pmxBnzzr3dtNTiWUhMpAg%40mail.gmail.com >> . >> For more options, visit https://groups.google.com/d/optout. >> > > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/c2ec459f-4a7b-497a-9586-7123a3d42399%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
