Re: Re: Re: Limit question
- To: mathgroup at smc.vnet.net
- Subject: [mg31554] Re: Re: Re: Limit question
- From: "Allan Hayes" <hay at haystack.demon.co.uk>
- Date: Sun, 11 Nov 2001 00:34:40 -0500 (EST)
- References: <9sigfu$avq$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Of course, Mathematica's does not add ComplexInfinity to the complex numbers
or +infinity and -infinity to the reals: the underlying notion is
DirectedInfinity.
FullForm[{Infinity, -Infinity, ComplexInfinity}]
List[DirectedInfinity[1],DirectedInfinity[-1],DirectedInfinity[]]
Geometrically we can project the complex plane onto a hemisphere touching at
zero, from the center of the corresponding sphere --- well, that does for
DirectedInfinity[z], but where does DirectedINfinity[] come in?
It does seem, as was pointed out earlier I think, that the default direction
might be -1:
Limit[1/x, x-> 0]//FullForm
DirectedInfinity[1]
Limit[1/x, x-> 0,Direction -> -1]//FullForm
DirectedInfinity[1]
Also, we have the expected
Limit[1/x, x-> 0, Direction->I]//FullForm
DirectedInfinity[Complex[0,1]]
But after
<<Calculus`Limit`
we get
Limit[1/x, x-> 0, Direction->I]//FullForm
DirectedInfinity[]
in spite of , for example,
Limit[1/x, x-> 0, Direction->-1]//FullForm
DirectedInfinity[1]
--
Allan
---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565
"Andrzej Kozlowski" <andrzej at tuins.ac.jp> wrote in message
news:9sigfu$avq$1 at smc.vnet.net...
> 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.
> >>>
> >>
> >>
> >
> >
>
>