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Re: Re: Re: "Assuming"
On 27 Feb 2008, at 00:19, Daniel Lichtblau wrote:
> Andrzej Kozlowski wrote:
>> On 26 Feb 2008, at 13:43, Daniel Lichtblau wrote:
>>> [...]
>>>
>>> I've seen cases where the FullSimplify[something] result differs
>>> from
>>> something on a finite set of integers. This motivated me several
>>> months
>>> ago to alter assumptions of integrality, to reality (realness?
>>> realhood?), in processing of Integrate.
>>>
>>> Daniel Lichtblau
>>> Wolfram Research
>>>
>> Formally speaking, reasonable measures (e.g. Radon measures) are
>> either diffuse or Dirac measures (or linear combinations of
>> these). However, for the former, the entire set of integers has
>> measure zero, and for the latter certain finite sets will have a
>> non-zero measure. So none of these seems to fit the intended
>> meaning of "measure zero".
>> More seriously; I think the intended meaning is that in
>> Simplify[thing1] -> thing2 thing1 and thing2 should both be
>> functions of some variable that is defined on an uncountable set,
>> then they may be are allowed to differ for a finite number of
>> values. But, if the functions are defined only on countable sets
>> (e.g. the set of all integers, as is the case with many number
>> theoretic functions) then the failure of thing2 to be equal to
>> thing1 on a finite set could be very serious. I think in such
>> situations the "set of measure zero" should really be the empty
>> set, or perhaps in really exceptional cases a set that contains no
>> more than a single point.
>> Andrzej Kozlowski
>
> Here is an example of the behavior in question. I do not pass
> judgement on whether it should be regarded as a bug or a feature. I
> simply wanted to give a concrete example where the behavior arises
> and is difficult to supress.
>
> In[2]:= i1 = Integrate[Sin[m*x]*Sin[n*x], {x,0,2*Pi}, Assumptions-
> >Element[{m,n},Reals]];
>
> Check what happens when we assign n->1 and then take limit as m->1.
>
> In[3]:= l1 = Limit[i1 /. n->1, m->1]
> Out[3]= Pi
>
> That was fine. Now see what happens if we assign n->1 and simplify
> under assumption that m is an arbitrary integer.
>
> In[4]:= l2 = Simplify[i1 /. n->1, Element[m,Integers]]
> Out[4]= 0
>
> Daniel Lichtblau
> Wolfram Research
>
>
Unless I am missing something obvious (which is possible as I have not
yet fully woken up) the problem amounts simply to this:
x = (1/2)*(Sin[2*(m - 1)*Pi]/(m - 1) -
Sin[2*(m + 1)*Pi]/(m + 1));
In[2]:= Limit[x, m -> 1]
Out[2]= Pi
In[3]:= Limit[x, m -> 1, Assumptions ->
Element[m, Integers]]
Out[3]= 0
The last answer maybe slightly dubious because it is not perfectly
clear in what sense the limit is taken here. But it seems to me a very
minor point and no cause for concern ?
Andrzej Kozlowski
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