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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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