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Re: Re: Re: Limit question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg31550] Re: [mg31528] Re: Re: Limit question
*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>
*Date*: Sat, 10 Nov 2001 01:19:39 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Well, this dispute is as much (or more) about mathematical conventions
as about mathematics. Formally speaking, to consider the concept of a
limit of a function or a mapping you have to have two topological
spaces X and Y and a mapping f: X->Y between them. In order to decide if
a limit exists or not one has to be clear about what the spaces X and Y
under consideration are. This of course is never explicitly clear in a
program like Mathematica. For example consider the function (mapping)
x -> 1/x. If you think of it as a mapping from the real line to the real
line than it it is reasonable ot say that limit[1/x,x->0] does not
exists and neither does limit[1/x^2,x->0] since the natural candidate
for such a limit, Infinity, is not a real number. In order to make sense
of such a concept it is common to adjoin to the real line two additional
"points" , + Infinity and -Infinity (topologically the real line now
becomes equivalent to a closed interval) and once one has done that one
can say that Limit[1/x^2, x->0] = + Infinity and Limit[-1/x^2,
x->0]=-Infinity. With this convention Limit[/x, x->0] still does not
exist. However, when considering the complex plane it is usual to add a
single point ComplexInfinity. With this point added the complex plane
becomes topologically a sphere (the Riemann sphere). The function x->1/x
nwo has a limit at 0, which is precisely ComplexInfinity. In fact it can
be extended to a holomorphic mapping from the Riemann sphere to itself
whose value at the point corresponding to 0 in the complex plane
(thought of as lying inside the sphere) is precisely ComplexInfinity.
When one says that Limit[1/x,x->0]= ComplexInfinity one is basically
referring to this sort of construction. Even in the real case one can do
something similar, by thinking of the function x ->1/x as taking values
not in the real line with two extra points (-Infinity, +Infinity), which
is topologically an interval, but in a topological circle which you
obtain by identifying these two infinities. Whn you use this convention
it is reasonable ot say that Limit[1/x, x->0] is just Infinity. In the
real case this is rarely useful and it is much more common to think of
+Infinity and -Infinity as distinct. In the complex case on the other
hand there is only one Infinity (ComplexInfinity) which has no direction
or rather all directions (all straight lines) lead to ComplexInfinity.
It's very difficult for a program like Mathematica to be consistent
about these things. In general Mathematica tries to interpret
Mathematical expressions in the widest possible context in which they
make sense. Thus a function like 1/x will normally be considered as a
complex valued function in the complex plane. However, when you use an
expression that only has a meaning in a restricted context (usually for
real numbers) Mathematica will restrict the domain of the expression to
real values. Thus the function UnitStep[x] *1/x will be considered as a
real function because it does not make sense to do otherwise. You can
sometimes force Mathematica to consider functions as real by inserting
UnitStep into their definition.
On quite many occasions however Mathematica (or rather the people
responsible for various aspects of the Mathematica kernel) seems to
make pretty ad hoc (not based on a general principle) judgments about
whether a certain expression should be considered as real or complex.
Most of the examples I know involve Integration or Limits. A somewhat
strange one is, in fact:
In[5]:=
Limit[1/x,x->0]
Out[5]=
Infinity
ComplexInfinity would have been more logical. It seems to me that
Mathematica is falling here in between the many stools it is trying to
sit on at once. Similar remarks apply to other cases that are from time
to time discussed on this list.
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
On Saturday, November 10, 2001, at 12:34 AM, Otto Linsuain wrote:
>
> I am confused by your remark. I think it is correct to say that
>
> Limit[1/x, x -> 0] and Limit[ Tan[x], x -> Pi/2]
>
> don't exist, even if x is allowed to go complex. I am not sure what you
> meant when you said that this is of course not true if one allows x to
> be
> a complex number. Perhaps you quoted the wrong piece of my message.
>
> Well, anyway, the bottom line is that Limit is pretty poorly implemented
> in Mathematica, although the improvements in the packages do help a lot.
>
> Otto Linsuain.
>
> On Fri, 9 Nov 2001, Andrzej Kozlowski wrote:
>
>> This is of course not true if you allow x to be a complex number, which
>> is exactly what Mathematica normally does in such cases.
>>
>> On Friday, November 9, 2001, at 08:13 PM, Otto Linsuain wrote:
>>
>>> As for the limits Limit[1/x, x->0] and Limit[Tan[x],x->Pi/2], without
>>> specifying a direction, the correct answer is that they don't exist.
>>>
>>
>>
>
>
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