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Re: Mathematica goes Bad

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59580] Re: Mathematica goes Bad
  • From: Maxim <ab_def at prontomail.com>
  • Date: Fri, 12 Aug 2005 03:37:57 -0400 (EDT)
  • References: <200508090730.DAA19089@smc.vnet.net> <ddcb4g$5bk$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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).

Also I think Mathematica often has problems with examples where the path  
along which the limit is taken or the path of integration isn't a straight  
line from the origin:

In[3]:= Integrate[(1 - Log[z])/z^2, {z, -1 - I, I*Infinity}]

Out[3]= (-(1/8) - I/8)*(3*Pi + 2*I*Log[2])

NIntegrate understands this as the integral along the line (-1 - I) + t*I  
with t running from 0 to Infinity, and Integrate works the same way, as  
can be seen from simpler examples. But then Out[3] is incorrect, because  
the antiderivative Log[z]/z is discontinuous on the path of integration,  
and the result is 2*Pi*I less than the correct value of the integral.

Maxim Rytin
m.r at inbox.ru


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