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Re: Re: Re: If Integrate returns no


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


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