On Tuesday, March 4, 2014 1:38:48 AM UTC-5, Richard Fateman wrote:   
   > On 3/3/2014 7:24 AM, hebisch@math.uni.wroc.pl wrote:   
   > > On Friday, February 28, 2014 11:33:56 PM UTC-5, Richard Fateman wrote:   
   > >> I don't consider a solution that includes   
   > >> Si, Ci, or hypergeometric functions as a solution   
   > >> in closed form in terms of elementary functions.   
   > >>   
   > >> Unless there is no way of expressing the answer in   
   > >> terms of elementary functions.   
   > >>   
   > >> After all, you could always decide that the   
   > >> difficult integral in question deserves its own   
   > >> name, say  FooI,  and then return the answer in terms   
   > >> of FooI.   
   > >   
   > >   _Indefinite_ integral above can not be done using elementary   
   > > functions.  For such integral 'li' and 'Ei' play the same   
   > > as logarithms.  'Ci' and 'Si' are similar to 'atan'.   
   > > As long as CAS can compute needed limits at infinity computing   
   > > indefinite integral in terms of special functions   
   > > is valid method of computing definite integral.   
   > > And it is much more general than methods based on   
   > > residue theorem.   
   >   
   > If your goal is not to compute a result in terms of elementary   
   > functions but to allow terms of the "higher" or "special" functions   
   > of applied mathematics, that's fine.   
      
   Ei has elementary asymptotics at 0 and at infinity, so _definite_   
   integral via limits freqenty will be elementary.  If Ei survives   
   in result of definte integration then almost surely there is no   
   elementary answer and then using Ei is better than nothing.   
   >   
   >   You just have to draw the   
   > line somewhere, e.g. functions in Abramowitz and Stegun. or   
   > the NIST digital library.   
      
   For integration of elementary function only Liouvilian functions   
   seem to help.  I am currently working on Ei/li, gamma incomplete   
   and polylogs.  For Ei/li and gamma incomplete there is extension   
   of Risch algorithm (extending Cherry).  This is still work in progress,   
   but one observation is that there is very little redundancy between   
   them.  Polylogs are more tricky, but there are some indications   
   that final theory may be as complete and as nice as of El/li.   
      
   Also, it would be nice to handle elliptic integrals. Nnd maybe   
   also abelian integrals on curves of higher genus, but   
   in case of higher genus I am not aware of established   
   canonical forms.   
      
   For differential equations it is important to handle MejerG,   
   so it is natural to allow it as part of integrands and results   
   of integration.   
      
   IMHO it is important that chosen function have little redundancy   
   and form reasonably complete system   
      
   > One of the uses of computer algebra systems is to find   
   > explicit formulas when possible, and it is a puzzle whether   
   > to use more functions, e.g. Si, Ci, Li;  or just express   
   > those and other functions as hypergeometric functions.   
      
   IMHO one should recognize important special cases, so basically have   
   MeijerG/hypergeometics in irreducible case, but simpler special forms   
   when they are Liouvilian/elementary.  In inner parts of CAS it   
   helps to have canonical forms, so Si/Ci may be reduced to Ei.   
   Cleary I want to avoid generating the same functions twice   
   under different names.  Users may want to use more forms,   
   and integration program should express (if possible) answer   
   in terms of functions appearing in users input.   
      
   --   
      
   Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
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