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Re: Re: Integration

On 9 Nov 2003, at 20:01, Alex wrote:

> Andrzej Kozlowski <akoz at> wrote in message 
> news:<boif1p$o9u$1 at>...
>> The former. This is what is meant by "in principle". There are lot's 
>> of
>> algorithms that work "in principle" but it is very easy to produce
>> fairly "simple looking" cases where no answer can be expected  during
>> the life-time of the user (or sometimes even mankind).
>> The fact that Mathematica does not arrive at an answer before the
>> user's patience is exhausted or his computer runs out of memory does
>> not mean, of course,  that Mathematica's implementation of an 
>> algorithm
>> is incomplete.
>> Andrzej Kozlowski
> For God's sake, of course one could imagine an integral which would
> take lifetime to compute. I am talking about mine which is rather
> simple and I computed it manually. How can one justify Mathematica not
> being able to handle it? Something is very bloody wrong with their
> implementation of their algorithm.
> Alex

I have not looked at your integral so I can't tell if you have computed 
it as you claim or not. But your claims that you can compute any 
integral and that the whole process is trivial  make me loose interest 
in even trying or having any discussions with you.
I have already mentioned Ramanujan, who is considered one of the 
greatest mathematical geniuses in history. One of his amazing skills  
was in computing indefinite integrals. That was of course long before 
the Risz algorithm (which in any case is generally impossible for 
humans to carry out). While I personally never compute any integrals by 
hand, I have a fair amount of experience in computing algebraic 
extensions (which as you should know by now is a part of the Risz 
algorithm) so I know that it is a far form trivial problem. (That is, 
by the way,  why I originally suggested you would have to be a 
reincarnation of Galois for me to take your claims seriously).

In fact, of course, as is stated on the MathWorld site, Mathematica's 
implementation of the algorithm is probably incomplete (so I wrong in 
writing that it wasn't) as is any other computer algebra 
implementation. It is precisely the difficulty of computing algebraic 
extensions that makes it so.
Personally I am not even convinced that Risz's the algorithm correct. 
You obviously seem to know a better one so you would do everyone a 
favor if you implemented it for us, ort at least described it as a 
sequence of implementable steps. Now that's a challenge worth taking 

Andrzej Kozlowski
Yokohama, Japan

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