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