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. > >>> > >> > >> > > > > > >