Re: Re: Mathematica goes Bad
- To: mathgroup at smc.vnet.net
- Subject: [mg59597] Re: [mg59580] Re: Mathematica goes Bad
- From: Andrzej Kozlowski <akozlowski at gmail.com>
- Date: Sat, 13 Aug 2005 03:26:31 -0400 (EDT)
- References: <200508090730.DAA19089@smc.vnet.net> <ddcb4g$5bk$1@smc.vnet.net> <200508120737.DAA17602@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 12 Aug 2005, at 09:37, Maxim wrote: > On Wed, 10 Aug 2005 07:42:08 +0000 (UTC), Daniel Lichtblau > <danl at wolfram.com> 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] -E^a while in the second Simplify[Exp[a + 2*I*Pi], a â?? Reals] E^a 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 uncontroversially 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] -Infinity Of course the current version of Mathematica returns Limit[ComplexExpand[E^(x + 2*I*ArcTan[x])], x -> Infinity] -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]] 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
- References:
- Mathematica goes Bad
- From: "Igor Touzov" <igor@nc.rr.com>
- Re: Mathematica goes Bad
- From: Maxim <ab_def@prontomail.com>
- Mathematica goes Bad