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Re: )

• To: mathgroup at smc.vnet.net
• Subject: [mg66313] Re: )
• From: Maxim <m.r at inbox.ru>
• Date: Tue, 9 May 2006 02:35:08 -0400 (EDT)
• References: <200605050902.FAA28575@smc.vnet.net> <e3hf84$m1h$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On Sat, 6 May 2006 06:20:52 +0000 (UTC), Andrzej Kozlowski  
<akoz at mimuw.edu.pl> wrote:

>
> laws of arithmetic do not hold). Some of them are hard to explain: I
> can't see any good reason at all why Infinity^Infinity is
> ComplexInfinity, and it seems to contradict the the most basic rule
> that x^y is always real when x and y are positive reals. Besides, as
> I mentioned earlier, Infinity and ComplexInfinity do not belong
> together in any topological model known to me (you need a
> "topological model" to be able to consider the issue of continuity)
> and should never appear in the same formula. I can only consider this
> as a bug, and a rather silly one.
>

I think this is as it should be: we need to consider all the sequences  
converging to Infinity, and, for example, Limit[(x + I*Pi)^x, x ->  
Infinity] == -Infinity. So in that sense Infinity^Infinity ==  
ComplexInfinity: when z = z1^z2 and z1, z2 go to Infinity the absolute  
value of z always tends to Infinity but the argument can be arbitrary.

Maxim Rytin
m.r at inbox.ru

• Follow-Ups:
  • Re: Re: )
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>

• References:
  • When is x^y = != E^(y*Log[x])
    • From: ted.ersek@tqci.net