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Re: Re: A Problem with Simplify


Alexey Popkov wrote:
> On 21 =C1=D0=D2, 11:26, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> 
>>And even in the purely algebraic cases  Reduce can easily take for
>>ever. Or consider this:
>>
>>  Reduce[x^3 + Sin[x] == 0, x]
>>
>>During evaluation of In[34]:= Reduce::"nsmet" :  "This system cannot
>>be solved with the methods available to Reduce"
>>
>>even though anyone can easily see that 0 is a solution (but Reduce is
>>not allowed to return an incomplete solution).
>>Adnrzej Kozlowski
> 
> 
> 
> I was surprised a bit. It is sad that even if I specify the Real
> domain I may not give the only possible answer x=0:
> Reduce[x + Sin[x] == 0, x, Reals]
> Solve[x + Sin[x] == 0, x, Reals]
> 
> Now I understand the depth of the problem.
> 
> But speaking about Integrate, is it really necessarily to perform
> Reduce[] on each step? The problem is to find the singularities on the
> parameters of the argument function (I mean such values of the
> parameters those degenerate the argument function).

Both integrand and indefinite integral might pose issues.


> After this we
> should keep track on arising new conditions on each step. It does not
> mean to use Reduce.

How else do you propose this might be done?


> We need only understand what we really do and know
> about limitations.

Here is a limitation. How does one figure out when, or where, a branch 
cut is crossed, if teh functions involved contain parameters?


> As I think this is not so much complicated task and
> may be fully implemented in Mathematica (if it is not implemented
> already).

Umm, not to put too fine a point on things, but quite clearly you have 
no experience with this. Suffice it to say that full analysis of 
parametrized singularities for definite integrals is not implemented in 
Mathematica already. Nor anywhere else. I suspect partial analysisof 
such to be better developed in Mathematica than in other software, but I 
cannot state that as a fact.


> On the final result we may need perform searching for the
> singularities again - but only for checking the result!
> 
> But as I see first of all Wolfram Research should extend Reduce[] for
> working with trigonometric functions.

Reduce[] handles trigs (alone) just fine. The example above mixes 
polynomial with trigonometric functions, of the same variable. handling 
this is quite nontrivial.


> This is that we should wait for
> nearest-future version of Mathematica. If we can not expect this -
> what for we should pay money?

I am not aware of any existing software that does what I believe you are 
requesting. That might give an indication that the algorithmic 
technology required is not yet out there.


Daniel Lichtblau
Wolfram Research


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