On Sun, Jan 03, 2021 at 04:05:48PM +0000, Dima Pasechnik wrote: > On Sat, Jan 2, 2021 at 4:28 PM Waldek Hebisch <[email protected]> > wrote: > > > > On Fri, Jan 01, 2021 at 08:15:42AM -0800, [email protected] wrote: > > > Has there been any progress on integrals expressible in elliptic > > > functions? > > > > > > I need them now, and it seems that no open source integrators can compute > > > things like > > > integrate(sqrt(x*(x-1)*(x-2)),x=0..1) in terms of elliptic functions. > > > > Nothing ready. I have a lot of distractions and on my side things > > go very slowly. > > Isn't integration of hypergeometric functions the right direction here?
Not exactly. For general elliptics one needs Lauricella D which ATM seem to increase difficulty. > > Integration is easy, identification of hypergeometrics doable, ATM it is not clear if identification of elliptics is computable. I hope it is solvable, but analogous problem for numbers (deciding if elliptic curve contains rational point) is deemed very hard. IME functions are easier than numbers, so identification _may_ turn out to be easy, but currently it is one of missing points (only heuristics are available). > what's > probably hardest is making > the results canonical - I am not sure how well the relations between > hypergeometric functions are > understood - on the other hand for 2F1 everything is known, and this > is the case of elliptic > integrals, right? IIUC 2F1 covers complete integrals (like your example). There are two main theoretical difficulties: identification of elliptics and simplifing relations between elliptics. For simplifying relations between elliptics there is quite strong theoretical result, but there is a catch: things are different for elliptic curves with complex multiplication and the ones without. AFAICS determining if curve has complex multiplication is doable using known methods, but computationally quite expensive. So one would like more efficient method. I can avoid checking for complex multiplication in some cases, but not always. Just as an explanation: classial works look at functions of single modulus. But there are some dependencies between functions of different moduli. This is analogous to dependence between exp(x)^2 and exp(2*x). Practialy, more important issue is dealing with algebraic extentions of constants. Some extentions are unavoidable, in particular when one wants to express elliptic in Jacobi form. But one can easily create unnecessay and complicated extentions. Your example is unusually easy, roots of defining polynomial are given explitiely and are integers. Apparently you want definite integrals. FriCAS usually handles definite integrals by doing indefinite integral first and then computing limits. So, assuming that we have handling of elliptics in indefinite integrator we would have to add handling of limits of elliptics (needed at least for complete integrals). Also, for definte integrals currently there is no hope for completeness: we get numbers as results and to determine independence of such numbers in general is beyond what current number theory can do. > > > > > On Tuesday, October 23, 2018 at 8:01:25 PM UTC+1 Waldek Hebisch wrote: > > > > > > > I looked a bit at FriCAS failures in Rubi testsutite. More than > > > > 8000 positions in testsutite contains elliptic integrals in > > > > answer. FriCAS currently can not generate ellipic integrals > > > > in answers, so unless the integral really is elementary > > > > (it happens sometimes, but is quite rare) FriCAS can not > > > > do it. This is more than 10% of the testsuite and single > > > > biggest reason for failures. > > > > > > > > More about this: it seems that most elliptic cases is > > > > very simple, easily reducing to defining formulas > > > > by few substitutions. It seem relatively easy to add > > > > ad-hoc handling for such cases. Main problem is that > > > > we do not want to loose completeness for elementary cases, > > > > so we can generate elliptics only after we decided that > > > > integral is nonelementary. > > > > > > > > Probably next biggest problem is polylogaritms (of order 5000 > > > > positions). We can handle one case, when argument of > > > > polylogarithm is an exponential. But Rubi testsuite seem > > > > to contain mostly different case. > > > > > > > > There are also integrals expressed in terms of hypergeometric > > > > functions (few thousends). They are used to integrate > > > > algebraic functions and some mixed cases involving > > > > algebraics and exponentials. I need to look closer, > > > > but at least some of them we should be able to handle > > > > like existing code for incomplete gamma. > > > > > > > > Together the cases above seem to cover vast majority of > > > > failures on Rubi testsuite. There are also failures > > > > which current FriCAS methods in principle should handle, > > > > but are not handled due to incomplete or buggy implementation. > > > > > > > > ATM better handling of the above is still in planning stage. > > > > I have made nice theoretical progress, both for elliptic > > > > functions and for polylogarithms. But in both cases > > > > theory is still too weak to give complete algorithm. > > > > So we probably should try adding ad-hoc extentions. > > > > There is reasonable chance that such extentions will > > > > be part of complete implementation in the future. > > > > > > > > -- > > > > Waldek Hebisch > > > > > > > > > > -- > > > You received this message because you are subscribed to the Google Groups > > > "FriCAS - computer algebra system" group. > > > To unsubscribe from this group and stop receiving emails from it, send an > > > email to [email protected]. > > > To view this discussion on the web visit > > > https://groups.google.com/d/msgid/fricas-devel/24a7f415-bec6-49ec-b98f-94eb188a88ecn%40googlegroups.com. > > > > > > -- > > Waldek Hebisch > > > > -- > > You received this message because you are subscribed to the Google Groups > > "FriCAS - computer algebra system" group. > > To unsubscribe from this group and stop receiving emails from it, send an > > email to [email protected]. > > To view this discussion on the web visit > > https://groups.google.com/d/msgid/fricas-devel/20210102162849.GA20948%40math.uni.wroc.pl. > > -- > You received this message because you are subscribed to the Google Groups > "FriCAS - computer algebra system" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/fricas-devel/CAAWYfq37fbAxd45CUL5ojCGNYX7KY7NasaTb6_m88YKw8NMo0g%40mail.gmail.com. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. 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