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Re: What is Infinity+Pi*I

ted.ersek at 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
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.

David W. Cantrell

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