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Re: What is Infinity+Pi*I

• To: mathgroup at smc.vnet.net
• Subject: [mg65883] Re: What is Infinity+Pi*I
• From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
• Date: Thu, 20 Apr 2006 05:15:20 -0400 (EDT)
• References: <e24ua4$5ql$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

ted.ersek at tqci.net wrote:
> I am using Mathematica 4.1, but I suspect all versions do the same in
> this case.
>
> In[1]:=
>      Infinity + Pi * I
>
> Out[1]=
>      Infinity
>
> I think it the input above should return itself.
> Am I wrong here? If we do it my way the following would return
>    -Infinity. (*Negative Infinity*)
>
> In[1]:=
>       E^( Infinity + Pi * I )
>
> Out[1]=
>       Infinity
>
> Either way it's an interesting example.

Yes. First, I recommend that you look at an extension of the complex plane
which was described in this newsgroup some time ago by Andrzej Kozlowski.
Please see
<http://groups.google.com/group/comp.soft-sys.math.mathematica/msg/ae824d2d32ae5a3d>
and possibly other parts of that thread. I had intended to
respond to AK in that thread, but never did. Anyway, here, from the draft
which I never posted, is part of what I was going to say to him:

"I like the model which you described. I've thought about that model
before, as, I suspect, have others. However, I'm not aware of its
having been discussed in the literature. (Reference anyone?) But
just as I don't expect to see Conway's surreal numbers implemented
in a CAS anytime soon, I don't suppose that the model of complex
infinities you described would be practical either. (BTW, I hope I'm
wrong about the issue of practicality. I would like to see that
model implemented.)"

If my pessimism about practicality of the above model is justified, perhaps
we should then ask: Of the simpler but less satisfactory models, which is
best? But I suspect that there is no clear answer to that question.

In Mathematica's model, Infinity + Pi*I simplifies to Infinity, as you
noted. And this causes Exp[Infinity + Pi*I] to be Infinity.
But of course Exp[Infinity]*Exp[Pi*I] is instead -Infinity, and so we have
a case where Exp[a + b] does not equal Exp[a]*Exp[b]. Regrettable. But
after all, Exp has an essential singularity at ComplexInfinity.

I suspect that any alternative models of comparable simplicity will also
exhibit some comparably regrettable behaviors.

Regards,
David W. Cantrell