Re: Re: Searching list for closest match to p
- To: mathgroup at smc.vnet.net
- Subject: [mg79558] Re: [mg79531] Re: Searching list for closest match to p
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sun, 29 Jul 2007 00:12:34 -0400 (EDT)
- References: <f81m5q$ll1$1@smc.vnet.net> <200707280942.FAA00210@smc.vnet.net> <803D65DC-3C02-4740-85D2-F56808D1007B@mimuw.edu.pl> <46ABA9FE.3010001@dordos.net>
On 28 Jul 2007, at 21:41, Peter Pein wrote: > Andrzej Kozlowski schrieb: > ... >> >> The OP statement of Jentzsch theorem was indeed inexact (as >> pointed out >> already by Daniel Lichtblau). The theorem says that for any power >> series >> in the complex plane, any point on the boundary of the disc of >> convergence is a limit of zero's of partial sums. >> >> There is a proof in the classic book of Titchmarsch "The Theory of >> Functions" (section 7.8). The proof takes a about 4 pages so I could >> scan it and send you but there is a snag: I only have a Russian >> translation of this book. So if you can read Russian I will send >> it to >> you (I think sending a scan of just 4 pages would not violate any >> copyright) but if not you will have to find someone who has the >> English >> original. >> >> Andrzej Kozlowski > > Dear Andrzej, > > there are many things on this wonderfull world which I do not > understand. The russian language is one of them. > > But with three universities in town it should be possible to find > this > book in a readable (for me) version. > > Thank you very much, > Peter > > P.S.: 4 pages... Iwould have thought, there is a short tricky and neat > way to show this, as it is true for many nice theorems of complex > analysis. My feeling says, it has to do with the fact that > somewhere on > the boundary has to be a singularity, but I did not think about > this yet. Russian is a very useful language to know as far as mathematics is concerned but I don't think it's worth learning it just to be able to read this ;-) (however, for me, just the ability to read numrous writings of my favorite mathematician of all time - V.I. Arnold, makes this language worth knowing) Actually, it is more like three pages, but as it begins near the end of a page it would be necessary to scan four. In fact, the proof in Titchmarsh's book is quite elementary and easty to understand, but it makes use of the well known theorem of Hurwitz, which states that given a (non-zero) analytic function f defined in a domain D whose boundary is a simple contour and which is a uniform limit of functions f[n] analytic on D, a point z in D is a root of f if and only if it is the limit of a sequence of roots of the functions f(n). So as usual in mathematics, the proof of Jentzsch theorem becomes "short" if you are willing to accept Horwitz theorem as granted. Hurwitz's theorem is proved a s a consequence to a theorem of Rouche, which in terms follows from the theorem of Cauchy ... What is "short" and "long" in mathemtics is quite often a matter of what you are willing to accept as given and not requiring proof. After all, a fully rigorous proof of the Jordan curve theorem has 6,500 lines and most people would be consider it as obvious! With best regards Andrzej
- References:
- Re: Searching list for closest match to p
- From: Peter Pein <petsie@dordos.net>
- Re: Searching list for closest match to p