I guess you are right. In modular forms one actually works with the
complex upper half plane
equipped with the action of a discrete group. If I am not mixing
things up (which is
quite likely as I am not an expert) you want a sort of minimal
compactification which is equivariant for the action
of this group. So infinity looks different....
Still in traditional one variable complex analysis I think it is quite
clear what "infinity" means
(which is what I defined it to be). The projective line over C is a
nice compact
complex manifold.
So the upshot is: if we want to speak formally about points at
infinity we need to specify
which compactification we use. In the case of the real line we have
the choice
of the the interval (-inf,+inf) and the circle (inf). The preferred
choice of compactification
depends on the context.
Michel
On Mar 28, 9:54 am, "William Stein" <[EMAIL PROTECTED]> wrote:
> On Fri, Mar 28, 2008 at 1:30 AM, Michel <[EMAIL PROTECTED]> wrote:
>
> > > Anyway, I'm not sure what to do about this. I don't even know what
> > > "complex infinity" means...
>
> > Sure you do (as someone working in modular forms)!
> > Infinity is the
> > point at infinity of the projective line over the
> > complex numbers (which is a 2-sphere). z<-->1/z exchanges complex
> > infinity and the origin.
> > That way you can speak about functions holomorphic at infinity etc...
>
> I meant that in the sense that the Maxima document
>
> "(%i8) describe(infinity);
> -- Constant: infinity
> `infinity' represents complex infinity."
>
> is far far too vague for me as a mathematician to be 100%
> certain of what it means. Basically I'm just very unsatisfied
> with (bad, imho) documentation like that.
>
> In particular, for me working in modular forms "infinity" is the
> point at infinity along the positive imaginary axis.
> There is nothing canonical about this though -- one could instead
> choose the negative imaginary axis if one wanted to. And
> this is evidently *not* what Maxima does, since:
>
> (%i2) limit(exp(2*%pi*%i*x), x, infinity);
> (%o2) und
>
> but lim_{x--> "oo"} exp(2*pi*i*x) = 0, where by "oo" I mean "my
> number theorist's infinity".
>
> This is also different than your definition above: "infinity is the
> point at infinity of the projective line over the
> complex numbers (which is a 2-sphere). z<-->1/z exchanges complex
> infinity and the origin." That doesn't really say what infinity is, but
> it implicitly defines limit_{z-->oo} f(z) to be lim_{z-->0} f(1/z), if
> it exists
> (in the sense of a limit of a complex variable, i.e., along any path to 0).
> With that definition, of course the limit lim_{x--> oo} exp(2*pi*i*x) is
> undefined.
>
> It would thus make a lot of sense that for Maxima
>
> limit(f(z), z, infinity)
>
> means limit(f(1/z), z, 0) as a complex function (which is undefined
> if the limit doesn't exist along all paths).
>
> Burcin writes:
> > Complex infinity is the unsigned infinity.
>
> I don't think that's helpful. It doesn't define what limit(f(z), z,
> infinity) means
> in Maxima. See above. I mean, certainly whatever Maxima means by
> "complex infinity" it should be unsigned. But I'm sure that's not what's
> meant by "unsigned_infinity" in Sage., i.e., I disagree that this
> is a bug in Sage:
>
> sage: maxima(unsigned_infinity)
> inf
>
> > Now we could also equip the complex plane with a "circle at
> > infinity". This is the Stone Cech compactification.
> > I guess this is just not the right thing for complex analysis.
>
> > Michel
>
> --
> William Stein
> Associate Professor of Mathematics
> University of Washingtonhttp://wstein.org
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