Re: Re: Re: If Integrate returns no

*To*: mathgroup at smc.vnet.net*Subject*: [mg87823] Re: [mg87807] Re: [mg87793] Re: [mg87759] If Integrate returns no*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Fri, 18 Apr 2008 02:38:34 -0400 (EDT)*References*: <200804161052.GAA29333@smc.vnet.net> <200804170235.WAA21166@smc.vnet.net> <200804171059.GAA15305@smc.vnet.net> <1E4F15AE-0C2D-485B-8ACD-6A2FB5CE831F@mimuw.edu.pl>

Andrzej Kozlowski wrote: > > On 17 Apr 2008, at 19:59, Matthias Bode wrote: > >> .. >> >>> "A completely implemented Risch algorithm will either return an >>> explicit answer for an integral that can be evaluated in terms of >>> elementary functions or determine that no such answer can be given." >>> ... >>> Andrzej Kozlowski >> >> >> Is this a theorem? >> >> Best regards, >> >> Matthias Bode. >> > > Yes. Actually quite many. The Risch's theorem, that I know, gives > conditions for an integral of a purely algebraic function to be > elementary and an algorithm for finding it or deciding that it does not > exist. There is also such an algorithm for a function which is an > element of the field K[t1,t2,...tn], where each tk is either an > exponential or a logarithm of a function in K[t1,t2,...,t(k-1)]. Again, > there is a theorem and an effective procedure. (However, even if fully > implemented these procedures may actually by impossible to carry out in > a reasonable time). But one can show, using them that things like > Integral[Log[Log[x]],x] cannot be expressed in terms of elementary > functions. Then, there is the mixed case, where you need both algebraic > and exponential or logarithmic extensions. For example, the function > ArcSin[x] belongs to a mixed extension, since > > TrigToExp[ArcSin[x]] > (-I)*Log[I*x + Sqrt[1 - x^2]] > > If I understand it correctly, there is also an effective procedure due > to Manuel Bronstein for dealing with general mixed cases (and it is > also a theorem) but it is much later work then Risch's (c.f. my > comments in a reply to a post by David Cantrell) and does not seem to > be implemented. In fact I am not sure about this since I have never > been interested in integrating non-algebraic functions so perhaps > someone else can confirm if I have got this right. > > Andrzej Kozlowski It might be implemented in AXIOM. I gather that much of Bronstein's work, as well as related indefinite integration algorithmics due to Barry Trager and James-not-John Davenport, is in AXIOM. Very likely implemented by those people themselves. As for Risch, what counts as "elementary" exp-log-powers is a bit tricky. Exp[x] and Log[x+1] count. Exp[1/2*Log[x+1]] does not count (it's algebraic). As best I understand, it is this last sort of thing for which work of Davenport, Trager, and Bronstein is required. Daniel

**References**:**If Integrate returns no result, can we conclude that no closed-form***From:*Szabolcs Horvát <szhorvat@gmail.com>

**Re: If Integrate returns no result, can we conclude that no closed-form***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: If Integrate returns no result, can we conclude that no closed-form***From:*"Matthias Bode" <lvsaba@hotmail.com>