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        riddles >> putnam exam (pure math) >> Convergence of double power series
        
(Message started by: Eigenray on Jul 15^th, 2005, 9:59pm)

Title: Convergence of double power series
Post by Eigenray on Jul 15^th, 2005, 9:59pm

Determine the open subsets U of the plane R² for which there exists a (complex) double power series [sum]_i,j>=0 a_ijzⁱw^j which (1) Converges for (log |z|, log |w|) in U, and (2) Diverges for (log |z|, log |w|) not in the closure of U.

One direction is, I think, fairly well-known, and holds for any number of variables. The other direction is more fun, but I don't know whether it's true in higher dimensions.

Title: Re: Convergence of double power series
Post by Eigenray on Aug 15^th, 2005, 1:01pm

Hrm...maybe I should give the result, and just ask for the proof? I think the most interesting part of the problem is to prove sufficiency: construct a power series for a given U.

In any event, I can say that it helps to consider some simple examples, and concern yourself only with absolute convergence at first.

Title: Re: Convergence of double power series
Post by Michael_Dagg on Aug 23^rd, 2005, 2:19pm

Check out the text Function Theory of Several Complex Variables by Krantz, in the first couple of chapters you can get an idea of how to show this by studying the material on logarithmically convex domains.

Title: Re: Convergence of double power series
Post by Eigenray on Aug 24^th, 2005, 8:43pm

Well I know the answer; this question was on my first problem set last year. If U is non-empty, then trivially it must contain, for some real r, {(x,y) | x,y < r}. That it must also be convex is not hard to show.

The heart of the problem is that these two conditions are also sufficient for the existence of a power series [sum] a_ijzⁱw^j converging on U and diverging outside its closure. The reader is cordially invited to give a construction demonstrating this.

Title: Re: Convergence of double power series
Post by Michael_Dagg on Nov 8^th, 2006, 6:03pm

By now of course what echos in my head is that
this is a standard characterization of those complete
Reinhardt domains that are the region of convergence of a
power series, so the answer should be that U is convex.

Title: Re: Convergence of double power series
Post by Eigenray on Dec 16^th, 2006, 1:22pm

The solution I have first considers the case that U is a half-plane.