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MathGroup Archive 2006

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Re: Re: Re: )


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On 13 May 2006, at 21:04, Andrzej Kozlowski wrote:

>  We have already seen that:
>
>
> Infinity^Infinity
>
> ComplexInfinity
>
> but note that actually:
>
> Limit[z^z, z -> Infinity]
>
> Infinity
>
> The choice of direction does not make any difference here:
>
>
> Limit[z^z, z -> Infinity, Direction -> I]
>
> Infinity
>
> Moreover:
>
>
> Limit[z^z, z -> ComplexInfinity]
>
> Infinity
>
> (I don't get this one)
>


Actually, I do get this one. It seems reasonable that Direction has  
no effect on

Limit[z^z, z -> Infinity]

since there is only straight line direction towards DirectedInfinity 
[1]. But, the situation is different in the case of ComplexInfinity.  
The seemingly strange answer

Limit[z^z, z ->ComplexInfinity]

Infinity

derives from the default direction of Limit (towards 1). So now I do  
get it now, although I still do not like it since I would much prefer  
the Riemann sphere model to be used consistently here and the answer  
ComplexInfinity to be returned. There is a certain duplication  
involved in the above answer, the two inputs (with one with Infinity  
and one with ComplexInfinity are interpreted by Mathematica to mean  
the same thing - because of the default direction).



But I can live with this.

This example is even  more interesting:

Limit[z^z, z -> ComplexInfinity, Direction->I]

0

This is completely reasonable, since


ComplexExpand[
   Abs[(a*I)^(a*I)]]


E^((-a)*Arg[I*a])

so the modulus tends to 0.

This, indeed agrees with

Limit[z^z,z->I Infinity]

0


So on the whole, I think limit already  works in a fairly reasonable  
way although I would still prefer and Assumptions based approach or  
maybe one based on an option Model, with values such as RiemannSphere  
and DirectedInfinities. In particular

Limit[1/z, z->0, Model->RiemannSphere] should return ComplexInfinity  
while
Limit[1/z,z->0,Model->DirectedInfinities] should return Infinity  
(which is what happens by default).

On the other hand, direct arithmetical operations on infinite  
quantities in mathematica,  still seem to me by and large meaningless.

Andrzej Kozlowski

Tokyo, Japan


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