Thanks a lot, John! (and sorry for the delay in answering)
I see the point: it is simply undefined.
But I do not quite agree with you in that "[one] would not expect a
computer algebra system to give the same answers with simple
substitutions in the two orders": ideally, I would expect that the
answer to substituting x=1 is something like "undefined", since it is
so. But I don't know if that is possible in Sage right now.
Cheers,
Jesús
2014-07-21 16:44 GMT+02:00 John Cremona <john.crem...@gmail.com>:
> The expression y = (1-x)/(1-x*cos(t)) is, as given, undefined whenever
> x*cos(t)=1, for example at (x,t)=(1,0).
>
> When x=1 it simplifies to 0/(1-cos(t)), which equals 0 except where
> cos(t)=1 where it is undefined but has a limiting value of 0.
>
> When t=0 it simplfies to (1-x)/(1-x), which equals 1 except when x=1
> where it is undefined, but has a limiting value of 1.
>
> So you get different limits when first x -> 1 and then t->0 compared
> with first t->0 and then x->1. The function has no continuous
> extension to (x,t)=(1,0). Hence I would not expect a computer algebra
> system to give the same answers with simple substitutions in the two
> orders.
>
> On 21 July 2014 15:14, kcrisman <kcris...@gmail.com> wrote:
>>> Hi guys,
>>>
>>> This is so simple that probably someone else has already noticed it, but
>>> just in case:
>>>
>>> sage: x,t = var('x,t')
>>> sage: f = (1-x)/(1-x*cos(t))
>>> sage: f(x=1)
>>> 0
>>> sage: f(t=0)(x=1)
>>> 1
>>>
>>
>> My guess is that this is more of a convention than anything else.
>>
>> sage: x/x
>> 1
>> sage: 0/x
>> 0
>>
>> Maxima:
>> (%i1) x/x;
>> (%o1) 1
>> (%i2) 0/x;
>> (%o2) 0
>>
>> If Mma and Maple do it too, that would be my guess. In any case, it is
>> 'known' and I bet you'll find other examples with a search of the email
>> lists (though searching for x/x might be hard!). It's possible to not
>> immediately do such reductions
>>
>> sage: x.mul(1/x,hold=True)
>> x/x
>>
>> but I'm not sure how to combine that with the substitution that you are
>> doing.
>>
>> - kcrisman
>>
>>> The second one is, of course, the correct answer. (FYI, Mathematica9
>>> fails, too.)
>>>
>>> Wouldn't the first one return some sort of conditional expression: "if t=0
>>> then 1, else 0"
>>>
>>> I would be happy to help in the debugging, if I can get some indication of
>>> what is running in the background, i.e. what function is called when one
>>> does the substitution "f(x=1)".
>>>
>>> Cheers,
>>> Jesús Torrado
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "sage-support" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> email to sage-support+unsubscr...@googlegroups.com.
>> To post to this group, send email to sage-support@googlegroups.com.
>> Visit this group at http://groups.google.com/group/sage-support.
>> For more options, visit https://groups.google.com/d/optout.
>
> --
> You received this message because you are subscribed to a topic in the Google
> Groups "sage-support" group.
> To unsubscribe from this topic, visit
> https://groups.google.com/d/topic/sage-support/Gxaui0CAzCM/unsubscribe.
> To unsubscribe from this group and all its topics, send an email to
> sage-support+unsubscr...@googlegroups.com.
> To post to this group, send email to sage-support@googlegroups.com.
> Visit this group at http://groups.google.com/group/sage-support.
> For more options, visit https://groups.google.com/d/optout.
--
You received this message because you are subscribed to the Google Groups
"sage-support" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-support+unsubscr...@googlegroups.com.
To post to this group, send email to sage-support@googlegroups.com.
Visit this group at http://groups.google.com/group/sage-support.
For more options, visit https://groups.google.com/d/optout.
