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Re: Re: Mathematica goes Bad

On 12 Aug 2005, at 09:37, Maxim wrote:

> On Wed, 10 Aug 2005 07:42:08 +0000 (UTC), Daniel Lichtblau
> <danl at> wrote:
>> I think it is safe to say that symbolic calculus is fraught with
>> problematic areas where, for example, mistakes involving a  
>> combination
>> of branch cuts and arithmetic with infinities can lead to erroneous
>> results. I do not see any evidence of this sort of phenomenon in the
>> example above, though. That was just a design decision. While you may
>> think it was the wrong thing to do, it is not symptomatic of deep
>> mathematical flaws.
>> I will note that by our metrics the number of open bugs in Limit  
>> dropped
>> sharply between versions 4 and 5 of Mathematica (I am not going to
>> quantify more closely than that).
>> Daniel Lichtblau
>> Wolfram Research
> Certainly this model of directed infinities has some peculiar  
> properties.
> For example, some functional identities don't hold when infinite
> quantities are involved:
> In[1]:= E^Infinity*E^(I*Pi) == E^(Infinity + I*Pi)
> Out[1]= False
> So E^(a + b) != E^a*E^b. This also means that Exp is not a continuous
> function anymore:
> In[2]:= Limit[E^(x + 2*I*ArcTan[x]), x -> Infinity]
> Out[2]= Infinity
> This is the result we get if we interchange Exp and Limit operations,
> which is equivalent to continuity. However, the correct result is
> -Infinity (Arg tends to Pi).

This is a good example, which indeed brings up one of the problematic  
aspects of Mathematica's "intuitive" and "informal" approach to  
certain mathematical issues. I am very fond of this informality, but  
I think that there are situations where it requires more care on the  
part of the user than would be the case with a program that required  
the user always to specify domains of functions, ground fields (real  
complex) for algebraic structures etc.

In particular,  when one considers such notions as of continuity and  
limits one has to be  clear about what topological space one is  
working working with. Leaving this out and using "intuition" can  
sometimes tie one in all sort of knots. Unfortunately, Mathematica's  
"intuitive" approach in this case easily leads to confusion. The  
problem is that Matheamtica allows one to combine together objects  
belonging to different mathematical realms and perform operations on  
them that only make sense in one of them.

One example of this is that Mathematica allows one to combine the  
three objects, Infinity, -Infinty and ComplexInfinity in the same  
expression. If one treats these as purely formal algebraic entities  
one can probably state a consistent set of rules for dealing with all  
three in this way, but it is a different matter if you want to  
consider topological notions such as convergence. You then need a  
well defined topological space and these three object do not belong  
to the same one.

In fact there are at least three topological spaces that are relevant  
in the context of Infinity, -Infinity and ComplexInfinity. First of  
all there is the standard two-point comactification of the real line  
(topologically the closed unit interval) [-Infinity,Infinity].  
Secondly, there is the "Riemann-sphere" one point compactification of  
the complex plane, consisting of all the complex numbers plus the  
point at infinity, denoted in Mathematica by ComplexInfinity.

Last  there is something problematic: another "compactification" of  
the complex plane  that turns it not into the Riemann sphere but into  
a closed disk, with the complex plane corresponding to the open disk  
whose boundary consists of DirectedInfinities in various directions.   
You can think of it as the complex numbers plus entities of the form   
DirectedInfinity[z], where z is a unit complex number. This  
compactification includes Infinity=DirectedInfinity[1]  and - 
Infinity=DirectedInfinity[-1] but not ComplexInfinity. It is this  
compactification that is often used by Mathematica. Unfortunately,  
this compactification has rather bad and unintuitive properties  
analytic properties.
One of them is that functions continuous on the complex plane  
(interior of the disc) may not have a continuous extension to the  
boundary. One such example is provided by the exponential function.  
Consider the simple question related to Maxim's example: what should  
be the value of Exp at DirectedInfinity[1]? One might think that it  
is enough to take a sequence of points in the complex plane  
converging to DirectedInfinity[1] and take the limit of this sequence  
as the value of Exp. But, of course this can't be done in the obvious  
way. Indeed, consider the sequences  x=a+ I Pi and x=a + 2 I Pi as a - 
 > DirectedInfinity[1] along the positive real line. In both cases x  
tends to DirectedInfinity[1] (although not along rays form 0!) but in  
the first case we have

Simplify[Exp[a + I*Pi], a â?? Reals]


while in the second

Simplify[Exp[a + 2*I*Pi], a â?? Reals]


Thus the first path should give us DirectedInfinity[-1] as the value  
of Exp[DirectedInfinity[1]] while the other should give  
DirectedInfinity[1]. This forces us to say that Exp[DirectedInfinity 
[1]] is actually not defined continuously althogh Exp[Infinity] is  
clearly Infinity. Here I am distinguishing Infinity (one of the ends  
of the two point compactification of the real line) and  
DirectedInfinity[1] but of course Mathematica considers them to be  
the same. Actually Exp[DirectedInfinity[1]] should either be left  
undefined  or defined as ComplexInfinity.

Of course if Mathematica consistently used the Riemann sphere one  
point compactification when dealing with complex infinities such  
problems would not occur. In the discussed example the argument would  
run as follows:

Consider the expression:

Limit[E^(x + 2*I*ArcTan[x]), x -> Infinity]

To start with, the fact that we see x->Infinity eliminates the  
possibility that x is a point on the Riemann sphere, since on the  
Riemann sphere no such thing as Infinity exists. So we must x as a  
real valued variable.  That means of course that 2*I*ArtTan[x] is now  
imaginary valued and x + 2*I*ArcTan[x] must lie on the Riemann  
sphere. We can now use continuity and the answer turns out to be  
ComplexInfinity (there is no Infinity or -Infinity in the Riemann  
sphere model). This would give ComplexInfinity as the final answer.

If we adopted this approach (Infinity and -Infinity are only used in  
a purely real context -- the two point compactification of the real  
line while in the complex context we always use ComplexInfinity) we  
could still get the more informative answer -Infinity but we would  
have to approach the problem differently. In order to avoid getting  
ComplexInfinity (which is the only infinity in the complex plane) we  
would have to make sure that the function whose limit we are seeking  
is always real valued. For example, in the above case we could do:

Limit[ComplexExpand[Re[E^(x + 2*I*ArcTan[x])]],
   x -> Infinity]


Of course the current version of Mathematica returns

Limit[ComplexExpand[E^(x + 2*I*ArcTan[x])],
   x -> Infinity]


which while happens to be what we wanted is, in my opinion,  
"incorrect", in view of what I tried to argue above. The correct  
answer ought to be ComplexInfinity. I realize that to many this may  
seem excessively pedantic but I can't see any other view to deal with  
the problem pointed out by Maxim. In other words the proposed  
solution is: return Infinity or -Infinity only as limits of functions  
which are unambiguously real valued; in all other cases return  
ComplexInfinity. DirectedInfinity[1] and DirectedInfinity[-1] should  
be distingushed from Infinity and -Infinity.
Expressions such as Infinity + I*Pi should either be returned  
unevaluated or interpreted as ComplexInfinity+I*Pi. One could still,  
use DirectedInfinity for limits:

Limit[x, x -> DirectedInfinity[I]]


although this answer should (probably) be ComplexInfinity.

The answer ComplexInfininity I consider correct in all cases, though  
of course to return it in all cases could sometimes be viewed a "loss  
of information".

If anyone has a better idea that solves the problem pointed out by  
Maxim then I would like to hear of it.

Andrzej Kozlowski

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