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 Washington
http://wstein.org
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