Re: Re: Searching list for closest match to p
- To: mathgroup at smc.vnet.net
- Subject: [mg79564] Re: [mg79531] Re: Searching list for closest match to p
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sun, 29 Jul 2007 00:15:40 -0400 (EDT)
- References: <f81m5q$ll1$1@smc.vnet.net> <200707280942.FAA00210@smc.vnet.net>
On 28 Jul 2007, at 10:42, Peter Pein wrote:
> chuck009 schrieb:
>> I'm working on an interesting theorem in Complex Analysis called
>> Jentzsch's Theorem in which the zeros of the partial sums of a
>> Taylor series for an analytic function in the unit disk, all
>> converge to values on the unit disk. So I choose a point on the
>> unit disk, p=Cos[pi/3]+iSin[pi/3], calculate the normal series for
>> f[x]=Log[1+x] for n ranging from 1 to 100, calculate the zeros,
>> then for each polynomial, search the zeros for the one closest to
>> the point. Here's my code. I feel the Table part is messy with
>> the First[First[Position... construct in it. Can anyone recommend
>> a more concise way of searching the zero lists and finding the one
>> closest to p3?
>>
>> Thanks,
>>
>>
>> p3color = Red;
>> p3 = Cos[Pi/3] + I*Sin[Pi/3];
>>
>> p3mintable =
>> Table[zlist = x /. N[Solve[Normal[Series[Log[1 + x], {x, 0,
>> nval}]] == 0],
>> 6]; minz = zlist[[First[First[Position[mins = (Abs[#1 -
>> p3] & ) /@
>> zlist, Min[mins]]]]]], {nval, 1, 100}];
>>
>> p3vals = ({Re[#1], Im[#1]} & ) /@ p3mintable;
>>
>> lp3 = ListPlot[p3vals, PlotRange -> {{-1.4, 1.4}, {-1.4, 1.4}},
>> AspectRatio -> 1];
>>
>> Show[{lp3, Graphics[{p3color, PointSize[0.03], Point[{Re[p3], Im
>> [p3]}]}],
>> Graphics[Circle[{0, 0}, 1]]}]
>>
> Dear Chuck,
>
> maybe you formulated inexact.
>
> I do not have Jentzsch's theorem at hand, but it would intuitively
> make
> more sense to me if: Let f be analytic *on C*. Then the zeros of the
> partial sums of the Taylor series of f all converge to values on the
> unit *circle*.
>
> f(0)=0 doesn't seem to matter.
>
> I didn't try to prove this (too lazy) but I've got an counterexample
> for your version:
> f(x)=x/(x-2)^2 is analytic on the unit disc, but:
> p3color=Red;
> circleColor=Blue;
> p3=Exp[I*Pi/3];
> f[x_]=x/(x-2)^2;
> s100=SeriesCoefficient[Series[f[x],{x,0,100}],#]x^#&/@Range[100];
>
> p3vals=Through[{Re,Im}[#]]&@@@
> Table[zlist=x/.NSolve[Total[Take[s100,nval]]\[Equal]0];
> Pick[zlist,#,Min[#]]&[Abs[zlist-p3]],{nval,100}];
>
> lp3=ListPlot[p3vals,PlotRange\[Rule]All,AspectRatio\[Rule]Automatic,
>
> Epilog\[Rule]{circleColor,Circle[{0,0},1,{0,Pi/2}],p3color,PointSize
> [.03],
> Point[Through[{Re,Im}[p3]]]}]
>
> shows that the zeroes do not converge to a z0 from the boundary of the
> unit disk.
> And constructing an easy case of a removable discontinuity with
> f[x_]=x/(Exp[x] - 1) shows, that the p3vals are all equal to zero.
> Using f[x_]=(x-Pi)/(Exp[x]-Exp[Pi]) to have a removable discontinuity
> outside D while being anlytic on D leads to Union[p3val] = {0} too.
>
> I have had a look at the net for this theorem (my books don't know it)
> but found only consequences. If you've got this in a common format
> (*.ps
> *.txt *.pdf or whatever) please send a copy to my email.
>
> Thanks in advance,
> Peter
>
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
- References:
- Re: Searching list for closest match to p
- From: Peter Pein <petsie@dordos.net>
- Re: Searching list for closest match to p