Re: Re: Re: If Integrate returns no result, can we conclude that no closed-form

*To*: mathgroup at smc.vnet.net*Subject*: [mg87831] Re: [mg87807] Re: [mg87793] Re: [mg87759] If Integrate returns no result, can we conclude that no closed-form*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Fri, 18 Apr 2008 02:40:01 -0400 (EDT)*References*: <200804161052.GAA29333@smc.vnet.net> <200804170235.WAA21166@smc.vnet.net> <200804171059.GAA15305@smc.vnet.net>

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

**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>