Author |
Topic: Convergence of double power series (Read 2152 times) |
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender: 
Posts: 1948
|
 |
Convergence of double power series
« on: Jul 15th, 2005, 9:59pm » |
 Quote  Modify
|
Determine the open subsets U of the plane R2 for which there exists a (complex) double power series [sum]i,j>=0 aijziwj 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.
|
| « Last Edit: Jul 15th, 2005, 10:15pm by Eigenray » |
 IP Logged |
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Convergence of double power series
« Reply #1 on: Aug 15th, 2005, 1:01pm » |
Quote Modify
|
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.
|
|
IP Logged |
|
|
|
Michael Dagg
Senior Riddler
Gender:
Posts: 500
|
|
Re: Convergence of double power series
« Reply #2 on: Aug 23rd, 2005, 2:19pm » |
Quote Modify
|
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.
|
|
IP Logged |
Regards,
Michael Dagg
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Convergence of double power series
« Reply #3 on: Aug 24th, 2005, 8:43pm » |
Quote Modify
|
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] aijziwj converging on U and diverging outside its closure. The reader is cordially invited to give a construction demonstrating this.
|
|
IP Logged |
|
|
|
Michael Dagg
Senior Riddler
Gender:
Posts: 500
|
|
Re: Convergence of double power series
« Reply #4 on: Nov 8th, 2006, 6:03pm » |
Quote Modify
|
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.
|
| « Last Edit: Nov 8th, 2006, 6:03pm by Michael Dagg » |
IP Logged |
Regards,
Michael Dagg
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Convergence of double power series
« Reply #5 on: Dec 16th, 2006, 1:22pm » |
Quote Modify
|
The solution I have first considers the case that U is a half-plane.
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print