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MathGroup Archive 2001

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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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