Re: Analytic functions of two variables

From: Chip Eastham (eastham_at_bellsouth.net)
Date: 08/21/04 

Date: Sat, 21 Aug 2004 13:27:39 -0400

"Lee Rudolph" <lrudolph@panix.com> wrote in message
news:cg7rqq$1do$1@panix2.panix.com...
> "Chip Eastham" <eastham@bellsouth.net> writes:
>
> >Some pointers, please, on how similar the theory of analytic functions in
> >two variables is to the single variable case.
>
> Do you refer to complex-analytic functions of two or more complex
variables?

I was thinking about exactly two, hoping to retain some geometric intuition.

> >In particular I'm wondering about how singularities might be distributed.
> >As a starting point, is it true that in a power series expansion of an
> >analytic function of two variables, say centered at the origin for
> >simplicity, the radius of convergence is up to the nearest singularity?
>
> What's "the radius of convergence", here? Are you assuming that
> (as in complex dimension 1) the interior of the domain on which a
> power series converges is an open (round) disk? That's false in
> dimension > 1; the right generalization is "polydisks" (maybe
> I mean "disc" and "polydisk"; I can never get that right),
> as a little experimentation with simple power series in two
> variables should convince you.

I'm convinced that the "radius" of convergence need not be "isotropic". I'd
hope first for a result that the domain of convergence is convex, and then
perhaps that it's an "ellipsoid" if construed in the right way.

> Leaving aside that kind of question, I think that the (fairly
> vast) dissimilarities between complex analysis in one or several
> variables are (at least "morally") all consequences of the simple
> fact that complex-analytic varieties of dimension 0 are just
> collections of isolated points, whereas a complex-analytic variety
> of dimension 1 or more always contains non-trivial open Riemann surfaces;
> isolated points don't themselves, support a whole lot of interesting
> analysis, so they don't impose *too* much of a restriction on (for
> instance) some analytic function of which they are all the singularities,
> but of course a non-trivial Riemann surface is chock-a-block with
> analytical (and topological) subtleties of its own, so it can make
> a big contribution to the subtlety of (for instance) an analytic
> function of which *it* belongs to the singularities. (Here, I'm
> using "singularities" in a sense which may not be the sense you're
> interested in, given your question about domains of convergence.
> But I think my comments probably still apply.)
>
> Lee Rudolph

Thanks for the thoughtful (as always) note & speed. (Doesn't Usenet ever
sleep?)

I suppose now I'm forced to divulge more of the motivation. I can remember
my surprise as a high school student on finding out that factorial, which
I'd always viewed as a essentially integer defined function, in fact has a
"continuous" extension via the gamma function. [More recently I was
surprised to discover the Windows calculator applet appears to do a nice job
evaluating fractional (real) factorials consistent with that extension.]

Now consider a function of two variables, the greatest common divisor. It
certainly seems to be intrinsically tied to the prime structure of the
integers, but we should no longer be surprised if it turned out to have a
useful complex extension. Of course we should be able to interpolate a
countable number of isolated points (I suspect, using a Blaschke (sp?)
product), but I'm looking for something "natural" (or perhaps more
difficult, a formulation of the nonexistence of a natural extension). [We
shall not speak of a complex version of pr*me c**nting functions, for fear
of awakening those who slumber.]

One clue seems to be gcd(z,z) = z, at least for z = 1,2,3,...

So maybe I'm looking for a "monodromy" theorem in two complex variables, to
get a feel for what makes an analytic continuation unique.

regards, Chip

Relevant Pages

  • Re: Analytic functions of two variables
    ... What's "the radius of convergence", ... fact that complex-analytic varieties of dimension 0 are just ... instance) some analytic function of which they are all the singularities, ... function of which *it* belongs to the singularities. ...
    (sci.math)
  • Re: Analytic functions of two variables
    ... >>In particular I'm wondering about how singularities might be distributed. ... the radius of convergence is up to the nearest singularity? ... >power series converges is an open disk? ... >instance) some analytic function of which they are all the singularities, ...
    (sci.math)
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