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

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,], 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

  (-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

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