
Re: integral for fun
Posted:
Mar 4, 2014 12:51 PM


On Tuesday, March 4, 2014 1:38:48 AM UTC5, 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 UTC5, 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

